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A telephone numbering plan is a type of numbering scheme used in telecommunications to allocate telephone numbers to subscribers and to route telephone calls in a telephone network. A closed numbering plan, such as found in North America, imposes a fixed total length to numbers. An open numbering plan features variance in the length of telephone numbers.

A dial plan establishes the expected number and pattern of digits for a telephone number. This includes country codes, access codes, area codes and all combinations of digits dialed. For instance, the North American public switched telephone network (PSTN) uses a 10-digit dial plan that includes a 3-digit area code and a 7-digit telephone number. Most PBXs support variable-length dial plans that use 3 to 11 digits. Dial plans must comply with the telephone networks to which they connect.

History[link]

In early telephone systems, connections were made in the central office by telephone operators using patch cords to connect one party to another. If a person wanted to make a phone call, in some phone systems he or she would pick up a phone and wind a crank. In other systems the person would wind the crank first, then pick up the phone. The crank was a small generator that would light a lamp at the central office. An operator would see the light and insert their patch cord into a socket and assist the customer with the call connection. The operator would use patch cords to connect the caller to the person being called. If the party being called was in another exchange, the operator would use a patch cord to connect to another exchange where an operator elsewhere would finish the connection. As technology advanced, electro-mechanical switches were introduced and calls were made using rotary dials.

Initial use of area codes in the United States and Canada began in 1947 with large cities.^[1] By 1966, the system was implemented fully in both countries.^[2] Area codes were assigned based on the length of time a rotary dial phone took to dial the area code. Densely populated areas like New York City, Chicago, Los Angeles, and Detroit had huge incoming call volume and were assigned numbers (212, 312, 213, 313) that could be quickly dialed from a rotary dial phone. On a rotary dial phone low digits (1, 2, 3, 4) could dial quickly as the time the rotary dial took to return to the home position was minimal. High digit numbers (7, 8, 9, 0) on rotary dial phones took much longer to return to the home position and were usually used in less densely populated areas like rural Texas (915). This numbering strategy became unnecessary when touch-tone phones arrived, as the tone allowed instant entry of digits.

This unreferenced section requires citations to ensure verifiability.

The original set of North American area codes was unique. When the North American Numbering Plan was first developed, logic circuits consisted of relays and vacuum tubes, and simplicity was mandatory. The second digit of all area codes was 0 or 1, while the second digit of the exchange triplet was never 0 or 1, thus facilitating readily the recognition whether the dialing of a full 10-digit number was in progress or whether the user was dialing only a local number. In this coding scheme, a leading 1 indicating long distance, or out of the area code call, was not necessary. In some regions, local numbers were 7 digits (occasionally less), but a toll call within the area code required a preceding “1”. This told the subscriber that the call being placed was a toll call. Area Code 617 in eastern Massachusetts used this system in the early 1970s, while the Chicago area (Area Code 312) did not. In the Chicago Area, one could call the Boston area by dialing only 10 digits, while in Boston, to call Chicago, one would be required to use 11 digits (preceding the 10 digits with a “1”)

By the 1990s, the mechanical Central Office Switches were rapidly being replaced with ESS machines (Electronic Switching Systems), and the Area Code logic was no longer necessary. The need for more telephone numbers was increasing rapidly, and the NANPA (North American Numbering Plan Administration) was running out of “n0n” and “n1n” combinations. This Area Code scheme was abandoned, with the result that former Area Code only numbers could now be Exchanges, and former Exchange only numbers could now be Area Codes. This caused a logic dilemma: The Switch had no way of knowing whether to expect 10 digits or 7 digits. The solution was simple. If a preceding “1” was entered (by dial and Touch-Tone key pad), then 10 more digits were expected. If the first digit entered was NOT a “1” only 7 digits were expected and the call would remain within area code. For a short while, in some area codes, one could enter the full 11 digits for a call within their own neighborhood or just enter the last 7 digits, and the call would be routed and billed identically.

Currently, because of Area Code overlays, nearly all metropolitan and many rural telephone calls require the full 11 digit entry to complete a telephone call.

Structure[link]

Most telephone numbers belong to the E.164 numbering plan, though some PABXs (business telephone systems) have internal extensions.

The E.164 numbering plan for telephone numbers includes:

• Country calling codes
• Regional numbering plans, such as:
  • the European Telephony Numbering Space
  • the North American Numbering Plan
• Various national numbering plans, such as:
  • Telephone numbers in the United Kingdom

Apart from the use of numbering plans for telephone numbers, they are also used in routing of SS7 signalling messages as part of the Global Title. In public land mobile networks, the E.212 numbering plan is used for subscriber identities (e.g. stored in the GSM SIM) while E.214 is used for routing database queries across PSTN networks.

Country code[link]

Country code - necessary only when dialing to phones in other countries. In international usage, telephone numbers are prepended with the country code preceded by a "+", and with spaces in place of hyphens (e.g., "+XX YYY ZZZ ZZZZ"). This allows the reader to choose which Access Code (also known as International Dialing Digit) they need to dial from their location. However, it is often quoted together with the international access code which must precede it in the dial string, for example "011" in NANP countries (including Canada, Bermuda, and the United States): "011-XX-YYY-ZZZ-ZZZZ", or "00" in most European countries: "00-XX-YYY-ZZZ-ZZZZ". This can cause confusion as a different Access Code may be used where the reader is located. On GSM networks, "+" is an actual character that may be used internally as the international access code, rather than simply being a convention.

Area code[link]

Area codes are also known as Numbering Plan Areas (NPAs) and formerly known as STD codes in the UK. These are necessary (for the most part) only when dialed from outside the code area, from mobile phones, and (especially within North America) from within overlay plans. Area codes usually indicate geographical areas within one country that are covered by perhaps hundreds of telephone exchanges, although the correlation to geographical area is becoming obsolete.^[3]

The area code is usually preceded in the dial string by either the national access code ("0" for many countries, "1" in USA and Canada) or the international access code and country code. However, this is not always the case, especially when 10-digit dialing is used. For example, in Montreal, Quebec, where area codes 514 and 450 are currently in use, users dial 10-digit number (e.g. 514 555 1234), dialing a 1 before this results in a recording advising not to dial a 1 as it is a local call. For non-geographic numbers, as well as mobile telephones outside of the North American Numbering Plan area, the "area code" does not correlate to a particular geographic area.

Area codes are often quoted including the national access code, for example a number in London: 020 8765 4321. Users must then correctly interpret the "020" as the code for London. If they call from another number in London, they merely dial 8765 4321, or if dialing from another country, drop the "0" and dial: +44 20 8877 9908.

000 area codes are sometimes used in the United States by telemarketers and VoIP services. They are only used for outgoing calls, and cannot be dialed.

Area codes were introduced in the United States by the Bell System, in 1947. The first direct dial long distance call occurred in 1951.^[4]

Local number[link]

The local number (or subscriber number) must always be dialed in its entirety. The first few digits in the local number typically indicate smaller geographical areas or individual telephone exchanges. In mobile networks they may indicate a network provider in case the area code does not. Callers from a number with a given area/country code usually do not need to (but optionally may) include the particular area/country code in the number dialed, which enables shorter "dial strings" to be used. Devices that dial phone numbers automatically can include the full number with area and access codes, since there is no additional annoyance related to dialing extra digits.

Standards[link]

Although the International Telecommunication Union (ITU) has attempted to promote common standards among nation states, numbering plans take different formats in different parts of the world. For example, the ITU recommends that member states adopt 00 as their international access code. However, as these recommendations are not binding on member states, some have not, such as the United States, Canada, and other countries and territories participating in the North American Numbering Plan.

The international numbering plan establishes country codes, that is, area codes that denote nations or groups of nations. The E.164 standard regulates country codes at the international level and sets a maximum length limit on a full international phone number (15 digits). However, it is each country's responsibility to define the numbering within its own network. As a result, regional area codes may be:

• A fixed length, e.g., three digits in the United States and Canada; two digits in Brazil; one digit in Australia and New Zealand,
• A variable length, e.g., between 2 and 5 in Germany, Argentina, United Kingdom and in Austria; between 1 and 5 in Japan; 1 or 2 in Syria and Peru, or
• Incorporated into the subscriber's number, as is the case in many countries, such as Spain or Norway. This is known as a "closed" telephone numbering plan. In some cases a trunk code (usually 0) must still be dialed, as in Belgium, Italy, Switzerland, South Africa and some locations within the NANP. Hong Kong used to have 1-digit area codes. They were incorporated into the subscriber's numbers back into the 1990s. No trunk code is required.

Area code and call pricing[link]

In countries other than the United States and Canada, the area codes generally determine the cost of a call. Calls within an area code and often a small group of adjacent or overlapping area codes are normally charged at a lower rate than outside the area code. This is not necessarily the case in the United States or Canada where rates are determined by the distance between rate centers. In most United States and Canadian locations area codes cover a sufficiently large territory that different rates will apply within the same area code determined by the distance between rate centers. Each rate center also has a local calling plan which determines which other rate centers, regardless of distance, are a local call. The area code plus the first three digits following the area code (the NPA-NXX) defines the rate center for any given telephone number. Multiple NPA-NXX combinations, often in different area codes, comprise each rate center, and each rate center is assigned a geographic point known as the rate center's V&H coordinates. The tariff distance in miles between any two rate centers may be calculated using by this formula: Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt { \frac{(V1-V2)^2 + (H1-H2)^2} {10} } .

Calls between nearby rate centers in different area codes may be cheaper (or even free local calls) as compared to calls to more distant rate centers in the same area code. Rates are set in zones of 0-6 mi, 6-12 mi, and so on, with these bands determined on a state-by-state basis for intrastate calls (calls within the same state) and determined by federal regulation for interstate calls (calls which cross a state line). As a specific example, callers in the Falls Church, Virginia, rate center (officially named "Washington Zone 17, VA"--example numbers beginning with 703-534, V=5636, H=1600) may make untimed local calls to 31 other nearby rate centers in Virginia, Maryland, and the District of Columbia in area codes 703, 571, 202, 301, and 240, while calls to distant locations in 703 such as Manassas and Haymarket, VA are charged as long distance.

Calls within a state [regulated by that state's public utilities commission] are often higher than rates to call more distant locations in some other state [regulated by the Federal Communications Commission]. The partial deregulation and introduction of competition for long distance phone services has established other methods of determining call pricing that do not necessarily follow the traditional model. Each year, more customers switch to a fixed rate, "all-you-can-dial" plan covering the state, the United States, or all North America generally (as of May 2008 and exclusive of taxes) for approximately $30 per month. Competition with cable telephony and Voice over Internet Protocol services have helped drive the cost of service down for residential and business customers.

Special area codes are generally used for free, premium rate, mobile phone systems (in countries where the mobile phone system is caller pays) and other special rate numbers. There are however some exceptions, in some countries (like Egypt), calls are charged at the same rate regardless of area and in others (like the UK) an area code is occasionally treated as two parts with different rates.

Dialing plans[link]

Example[link]

In the U.S., some typical dial plans include:

• Internal extension numbers of a varying number of digits
• Local numbers of seven or ten digits
• Long distance numbers of eleven digits, consisting of a 1, then a three-digit area code, then a seven-digit number
• International numbers of any length starting with 011

Many corporate telephone systems use a dialing prefix, typically the digit 9, to obtain a line to the public switched telephone network.

Example[link]

In Australia, the following dialing plans can be found:

• Internal extension numbers of a varying number of digits, usually 3 or 4 digits
• Local numbers of eight digits
• Long distance numbers of eight digits, when source and destination share the same area code, or ten digits, consisting of a 0, then an area code, then eight-digit number
• International numbers of any length starting with 0011

In Australia, the digit 0 is used to obtain a line to the public switched telephone network.

Open dialing plans[link]

An open dialing plan is one in which there are different dialing arrangements for local and long distance telephone calls. This means that to call another number within the same city or area, callers need dial only the number, but for calls outside the area, an area code is required. In this situation it is customary to show the area code in parentheses, signifying that in some cases the area code is optional or is not required, as suggested by ITU-T Recommendation E.123. The area code is prefixed by a trunk code (usually "0"), which (apart from rare cases such as Italian land lines) is omitted when calling from outside the country.

To call a number in Sydney, Australia for example:

• xxxx xxxx (within Sydney and other locations within New South Wales - no area code required)
• (02) xxxx xxxx (outside New South Wales - the area code is required)
• +61 2 xxxx xxxx (outside Australia; notice the dropped 0)

Note that the "+" is not dialed, it signifies that first the international access code must be dialed, followed by the country code, in this case 61, followed by the number. When using a mobile telephone to place a call, all GSM phones and many models using other technologies do allow the + to be entered and this is internally converted to the correct access code, based on caller location, as the call is made.

New Zealand has a special case of an open dialing plan. While most nations require the area code to be dialed only if it is different, in New Zealand, one needs to dial the area code if the phone is outside the local calling area. For example, the town of Waikouaiti is in the Dunedin City Council jurisdiction, and has phone numbers (03) 465 7xxx. To call the city council in central Dunedin (03) 477 4000, residents must dial the number in full including the area code even though the area code is the same, as Waikouaiti and Dunedin lie in different local calling areas (Palmerston and Dunedin respectively)^[5]

In the United States, Canada, and other countries or territories using the North American Numbering Plan (NANP), the trunk code is '1', which is also (by coincidence) the country calling code. The same rule also applies in many parts of the NANP, including all areas of Canada that still have an open dialing plan. This is not universal, as there are locations within the United States that allow long distance calls within the same area code to be dialed as seven digits. In Canada, the trunk code (also known as the long distance access code) must also be dialed along with the area code for long distance calls even within the same area code. For example, to call a number in Regina:

• xxx xxxx (within Regina, Lumsden and other local areas)
• 1 (306) xxx xxxx (within the 306 area code, but not within the Regina local calling area (e.g., Saskatoon))
• 1 (306) xxx xxxx (anywhere within the NANP outside the 306 area code)
• +1 (306) xxx xxxx (outside NANP)

To call a number in San Francisco, California the dialing procedure will vary:

• xxx xxxx (local calls, no area code required)
• 1 (415) xxx xxxx (outside San Francisco, and within the U.S., Canada, and other countries in the NANP)^[6]
• +1 (415) xxx xxxx (outside the NANP - 1 is the country code for the U.S.)

However, in parts of North America, especially where a new area code overlays an older area code, dialing the area code, or 1 + the area code, is now required even for local calls, which means that the NANP is now closed in certain areas and open in others. Dialing from mobile phones is different in the U.S., as the trunk code is not necessary, although it is still necessary for calling all long distance numbers from a mobile phone in Canada. (Most mobile phones today can be programmed to automatically add a frequently-called area code as a prefix, allowing calls within the desired area to be dialed by the user as seven-digit numbers, though sent by the phone as 10-digit numbers.)

In some parts of the United States, especially northeastern states such as Pennsylvania served by Verizon Communications, the full 10-digit number must be dialed. If the call is not local, the call will not complete unless the dialed number is preceded by a 1. In this situation, where the area code is not optional, the area code is not enclosed in parentheses. Thus:

• 610 xxx xxxx (local calls, area code required; one of two completion options for mobile phones within the U.S.)
• 1 610 xxx xxxx (intra-area code charged calls; second of two completion options for mobile phones within the U.S.)
• +1 610 xxx xxxx (outside the NANP)

In California, because of the existence of both Overlay Area Codes (where you must dial an area code with every call regardless) and non-Overlay area codes (where you only have to dial an area code for calls outside your 'home' area code), "permissive home area code dialing" of 1 + the area code within the same area code, even if no area code is required, has been permitted since the mid-2000's. e.g.: if you are the 213 area code (a non-overlay area code), you can dial calls as 7 digits (XXX-XXXX) or 1-213 + 7 digits. The manner in which you dial the call does not affect the cost of the call. This "permissive home area code dialing" helps maintain uniformity and eliminates confusion given the different types of area code relief that has made California the nation’s most "area code" intensive State. Unfortunately, unlike other States with Overlay Area codes (Texas, Maryland, Florida and Pennsylvania to name just a few), the California Public Utilities Commission maintains two different dialing plans: Landlines must dial 1 + area code whenever an Area Code is part of the dialed digits while Cell Phone can omit the "1" and just dial 10 digits.

Many organizations have private branch exchange systems which permit dialing the access digit(s) for an outside line (usually 9 or 8), a "1" and finally the local area code and xxx xxxx in areas without overlays. This "feature" is unintentionally helpful for employees who reside in one area code and work in an area code with one, two, or three adjacent area codes. "1+" dialing to any area code by an employee can be done quickly, with all exceptions processed by the private branch exchange and passed onto the public switched telephone network.

Open and closed dialing plans should not be confused with open and closed numbering plans. A closed numbering plan, such as found in North America, features fixed length area codes and local numbers. An open numbering plan, as found in assorted countries that have not yet standardized, features variance in length of area code or local number, or both. Closed dialing plans are rare where numbering plans are open.

Closed dialing plans[link]

A closed dialing plan is one in which the subscriber's full number is used for all calls, even in the same area. This has traditionally been the case in small countries and territories where area codes have not been required. However, there has been a trend in many countries towards making all numbers a standard length, and incorporating the area code into the subscriber's number. This usually makes the use of a trunk code obsolete. For example, to call Oslo in Norway before 1992, one would dial:

• xxx xxx (within Oslo - no area code required)
• (02) xxx xxx (within Norway - outside Oslo)
• +47 2 xxx xxx (outside Norway)

After 1992, this changed to a closed eight-digit numbering plan, e.g.:

• 22xx xxxx (within Norway - including Oslo)
• +47 22xx xxxx (outside Norway)

Therefore in other countries, such as France, Belgium, Switzerland, South Africa and some parts of North America where the dialing plan is closed, the trunk code is retained for domestic calls, whether local or national, e.g.,

• Paris 01 xx xx xx xx (outside France +33 1 xxxx xxxx)
• Brussels 02 xxx xxxx (outside Belgium +32 2 xxx xxxx)
• Geneva 022 xxx xxxx (outside Switzerland +41 22 xxx xxxx)
• Cape Town 021 xxx xxxx (outside South Africa +27 21 xxx xxxx)
• New York 1 212 xxx xxxx (outside the North American Numbering Plan +1 212 xxx xxxx)

while some, like Italy, require the initial zero to be dialled, even for calls from outside the country, e.g.,

• Rome 06 xxxxxxxx (outside Italy +39 06 xxxxxxxx)

Further, there are locations with closed dialing plans in the NANP that require the full phone number including area code to be dialed for all calls, but the trunk code is required for only long distance calls, even in the same area code.

While the use of full national dialing is less user-friendly than using only a local number without the area code, the increased use of mobile phones, which can store numbers, means that this is of decreasing importance. It also makes easier to display numbers in the international format, as no trunk code is required—hence a number in Prague, Czech Republic, can now be displayed as:

• 2xx xxx xxx (inside Czech Republic)
• +420 2xx xxx xxx (outside Czech Republic)

as opposed to before September 21, 2002:^[7]

• 02 / xx xx xx xx (inside Czech Republic)
• +420 2 / xx xx xx xx (outside Czech Republic)

Some countries already switched, but trunk prefix re-added with the closed dialing plan, for example in Bangkok, Thailand before 1997:

• xxx-xxxx (inside Bangkok)
• 02-xxx-xxxx (inside Thailand)
• +66 2-xxx-xxxx (outside Thailand)

has been switched in 1997:

• 2-xxx-xxxx (inside Thailand)
• +66 2-xxx-xxxx (outside Thailand)

Trunk prefix has re-added in 2001

• 02-xxx-xxxx (inside Thailand)
• +66 2-xxx-xxxx (outside Thailand)

Satellite phone numbering plans[link]

Satellite phones are usually issued with numbers in a special country calling code. For example, Inmarsat satellite phones are issued with code +870, while Global Mobile Satellite System providers, such as Iridium, issue numbers in country code +881 ("Global Mobile Satellite System") or +882 ("International Networks"). Some satellite phones are issued with ordinary phone numbers, such as Globalstar satellite phones issued with NANP telephone numbers.

• Inmarsat:+870: SNAC (Single Network Access Code)
• ICO Global: +881 0, +881 1.
• Ellipso: +881 2, +881 3.
• Iridium: +881 6, +881 7.
• Globalstar: +881 8, +881 9.
• Emsat: +882 13.
• Thuraya: +882 16.
• ACeS: +882 20.

+ 88184

Special services[link]

Some country calling codes are issued for special services, or for international/inter regional zones.

• +388 5 – shared code for groups of nations
• +388 3 – European Telephony Numbering Space – Europe-wide services (discontinued)
• +800 – International Freephone (UIFN)
• +808 – reserved for Shared Cost Services
• +878 – Universal Personal Telecommunications services
• +881 – Global Mobile Satellite System
• +882 – International Networks
• +888 - international disaster relief operations
• +979 – International Premium Rate Service
• +991 – International Telecommunications Public Correspondence Service trial (ITPCS)
• +996 9 - Kyrgyzstan PG.R.S. Server
• +999 – reserved for future global service

Numbering plans by global regions[link]

• Telephone numbers in the Americas
• Telephone numbers in Oceania
• Telephone numbers in Europe
• Telephone numbers in Asia
• Telephone numbers in Africa

Dial plan syntax[link]

The following syntax used to identify a dial plan in a digit map is adapted from [RFC 2705]^[8].

To specify a

Enter the following

Result

Some dial plan examples using the above syntax look as follows:

For calls to

Users dial

Dial plan

Numbering plan indicator[link]

The numbering plan indicator (NPI) is a number which is defined in the ITU standard Q.713, paragraph 3.4.2.3.3, indicating the numbering plan of the attached telephone number. NPIs can be found in SCCP and SMS messages. As of 2004^[update], the following numbering plans and their respective numbering plan indicator values have been defined:

See also[link]

• List of North American Numbering Plan area codes for the U.S., Canada, Caribbean, and Pacific Ocean islands
• List of dialling codes in the United Kingdom
• Telephone numbers in Australia
• Carrier access code

External links[link]

• http://www.itu.int/dms_pub/itu-t/opb/sp/T-SP-LT.1-2011-PDF-E.pdf
• List of ITU-T Recommendation E.164 assigned country codes as of 1 Nov 2011
• http://www.itu.int/dms_pub/itu-t/opb/sp/T-SP-E.164C-2011-PDF-E.pdf of ITU-T Recommendation E.164 Dialling Procedures as of 15 DEC 2011]
• World Telephone Numbering Guide

References[link]

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The combined area of these three shapes is between 15 and 16 squares.

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (m²), which is the area of a square whose sides are one metre long.^[1] A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles.^[2] For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.^[3]

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.^[4] In analysis, the area of a subset of the plane is defined using Lebesgue measure,^[5] though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

Formal definition[link]

An approach to defining what is meant by area is through axioms. For example, we may define area as a function a from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers which satisfies the following properties:

• For all S in M, a(S) ≥ 0.
• If S and T are in M then so are S ∪ T and S ∩ T, and also a(S∪T) = a(S) + a(T) − a(S∩T).
• If S and T are in M with S ⊆ T then T − S is in M and a(T−S) = a(T) − a(S).
• If a set S is in M and S is congruent to T then T is also in M and a(S) = a(T).
• Every rectangle R is in M. If the rectangle has length h and breadth k then a(R) = hk.
• Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a(S) ≤ c ≤ a(T) for all such step regions S and T, then a(Q) = c.

It can be proved that such an area function actually exists. (See, for example, Elementary Geometry from an Advanced Standpoint by Edwin Moise.)

Units[link]

A square metre quadrat made of PVC pipe.

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measure in square metres (m²), square centimetres (cm²), square millimetres (mm²), square kilometres (km²), square feet (ft²), square yards (yd²), square miles (mi²), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.

The SI unit of area is the square metre, which is considered an SI derived unit.

Conversions[link]

Although there are 10 mm in 1 cm, there are 100 mm² in 1 cm².

The conversion between two square units is the square of the conversion between the corresponding length units. For example, since

1 foot = 12 inches,

the relationship between square feet and square inches is

1 square foot = 144 square inches,

where 144 = 12² = 12 × 12. Similarly:

• 1 square kilometer = 1,000,000 square meters
• 1 square meter = 10,000 square centimetres = 1,000,000 square millimetres
• 1 square centimetre = 100 square millimetres
• 1 square yard = 9 square feet
• 1 square mile = 3,097,600 square yards = 27,878,400 square feet

In addition,

• 1 square inch = 6.4516 square centimetres
• 1 square foot = 0.09290304 square metres
• 1 square yard = 0.83612736 square metres
• 1 square mile = 2.589988110336 square kilometres

Other units[link]

There are several other common units for area. The are was the original unit of area in the metric system, with

• 1 are = 100 square metres

Though the are has fallen out of use, the hectare is still commonly used to measure land:

• 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres

Other uncommon metric units of area include the tetrad, the hectad, and the myriad.

The acre is also commonly used to measure land areas, where

• 1 acre = 4,840 square yards = 43,560 square feet.

An acre is approximately 40% of a hectare.

On the atomic scale, area is measured in units of barns, such that,

• 1 barn = 10⁻²⁸ square meters.

The barn is commonly used in describing the cross sectional area of interaction in nuclear physics.

Basic area formula[link]

Rectangles[link]

The area of this rectangle is lw.

The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length l and w, the formula for the area is

A = lw  (rectangle).

That is, the area of the rectangle is the length multiplied by the width. As a special case, the area of a square with side length s is given by the formula

A = s²  (square).

The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers.

Equal area figures.

Dissection formulae[link]

Most other simple formulae for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.

For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:

A = bh  (parallelogram).

Two equal triangles.

However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram:

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{1}{2}bh

 (triangle).

Similar arguments can be used to find area formulae for the trapezoid and the rhombus, as well as more complicated polygons.

Circles[link]

A circle can be divided into sectors which rearrange to form an approximate parallelogram.

The formula for the area of a circle is based on a similar method. Given a circle of radius r, it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is r, and the width is half the circumference of the circle, or πr. Thus, the total area of the circle is r × πr, or πr²:

A = πr²  (circle).

Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly πr², which is the area of the circle.

This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \;=\; \int_{-r}^r 2\sqrt{r^2 - x^2}\,dx \;=\; \pi r^2.

Surface area[link]

Archimedes showed that the surface area and volume of a sphere is exactly 2/3 of the area and volume of the surrounding cylindrical surface.

Most basic formulae for surface area can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a cylinder (or any prism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone, the side surface can be flattened out into a sector of a circle, and the resulting area computed.

The formula for the surface area of a sphere is more difficult: because the surface of a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is

A = 4πr²  (sphere).

where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus.

List of formulae[link]

is the length of one side of the triangle.

is half the perimeter, Failed to parse (Missing texvc executable; please see math/README to configure.): a

and Failed to parse (Missing texvc executable; please see math/README to configure.): c
are the length of each side.

and Failed to parse (Missing texvc executable; please see math/README to configure.): b
are any two sides, and Failed to parse (Missing texvc executable; please see math/README to configure.): C
is the angle between them.

and Failed to parse (Missing texvc executable; please see math/README to configure.): h
are the base and altitude (measured perpendicular to the base), respectively.

is the length of one side of the square.

and Failed to parse (Missing texvc executable; please see math/README to configure.): w
are the lengths of the rectangle's sides (length and width).

and Failed to parse (Missing texvc executable; please see math/README to configure.): b
are the lengths of the two diagonals of the rhombus.

is the length of the base and Failed to parse (Missing texvc executable; please see math/README to configure.): h
is the perpendicular height.

and Failed to parse (Missing texvc executable; please see math/README to configure.): b
are the parallel sides and Failed to parse (Missing texvc executable; please see math/README to configure.): h
the distance (height) between the parallels.

is the length of one side of the hexagon.

is the length of one side of the octagon.

     is the side length and Failed to parse (Missing texvc executable; please see math/README to configure.): n
is the number of sides.

     is the perimeter and Failed to parse (Missing texvc executable; please see math/README to configure.): n
is the number of sides.

     is the radius of a circumscribed circle, Failed to parse (Missing texvc executable; please see math/README to configure.): r
is the radius of an inscribed circle, and Failed to parse (Missing texvc executable; please see math/README to configure.): n
is the number of sides.

is the apothem, or the radius of an inscribed circle in the polygon, and Failed to parse (Missing texvc executable; please see math/README to configure.): p
is the perimeter of the polygon.

is the radius and Failed to parse (Missing texvc executable; please see math/README to configure.): d
the diameter.

and Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
are the radius and angle (in radians), respectively.

and Failed to parse (Missing texvc executable; please see math/README to configure.): b
are the semi-major and semi-minor axes, respectively.

and Failed to parse (Missing texvc executable; please see math/README to configure.): h
are the radius and height, respectively.

and Failed to parse (Missing texvc executable; please see math/README to configure.): h
are the radius and height, respectively.

and Failed to parse (Missing texvc executable; please see math/README to configure.): l
are the radius and slant height, respectively.

and Failed to parse (Missing texvc executable; please see math/README to configure.): l
are the radius and slant height, respectively.

and Failed to parse (Missing texvc executable; please see math/README to configure.): d
are the radius and diameter, respectively.

is the base area, Failed to parse (Missing texvc executable; please see math/README to configure.): P
is the base perimeter and Failed to parse (Missing texvc executable; please see math/README to configure.): L
is the slant height.

is the area of the square in square units.

is the area of the circle in circular units.

The above calculations show how to find the area of many common shapes.

The area of irregular polygons can be calculated using the "Surveyor's formula".^[6]

Additional formulae[link]

Areas of 2-dimensional figures[link]

• a triangle: Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac12Bh

(where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then Heron's formula can be used: Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{s(s-a)(s-b)(s-c)}
where a, b, c are the sides of the triangle, and Failed to parse (Missing texvc executable; please see math/README to configure.): s = \tfrac12(a + b + c)
is half of its perimeter. If an angle and its two included sides are given, the area is Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac12 a b \sin(C)
where C is the given angle and a and b are its included sides.  If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac12(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3)

. This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points (x₁,y₁), (x₂,y₂), and (x₃,y₃). The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use Infinitesimal calculus to find the area.

• a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: Failed to parse (Missing texvc executable; please see math/README to configure.): i + \frac{b}{2} - 1

, where i is the number of grid points inside the polygon and b is the number of boundary points. This result is known as Pick's theorem.

Area in calculus[link]

Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).

The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

• The area between a positive-valued curve and the horizontal axis, measured between two values a and b (b>a) on the horizontal axis, is given by the integral from a to b of the function that represents the curve.
• The area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x).
• An area bounded by a function r = r(θ) expressed in polar coordinates is Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over 2} \int_0^{2\pi} r^2 \, d\theta

.

• The area enclosed by a parametric curve Failed to parse (Missing texvc executable; please see math/README to configure.): \vec u(t) = (x(t), y(t))

with endpoints Failed to parse (Missing texvc executable; please see math/README to configure.):  \vec u(t_0) = \vec u(t_1) 
is given by the line integrals

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_{t_0}^{t_1} x \dot y \, dt = - \oint_{t_0}^{t_1} y \dot x \, dt = {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt

(see Green's theorem) or the z-component of

Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt.

Surface area of 3-dimensional figures[link]

• cone: Failed to parse (Missing texvc executable; please see math/README to configure.): \pi r\left(r + \sqrt{r^2 + h^2}\right)

, where r is the radius of the circular base, and h is the height. That can also be rewritten as Failed to parse (Missing texvc executable; please see math/README to configure.): \pi r^2 + \pi r l

where r is the radius and l is the slant height of the cone. Failed to parse (Missing texvc executable; please see math/README to configure.): \pi r^2 
is the base area while Failed to parse (Missing texvc executable; please see math/README to configure.): \pi r l 
is the lateral surface area of the cone.

• cube: Failed to parse (Missing texvc executable; please see math/README to configure.): 6s^2

, where s is the length of an edge.

• cylinder: Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi r(r + h)

, where r is the radius of a base and h is the height. The 2Failed to parse (Missing texvc executable; please see math/README to configure.): \pi r can also be rewritten as Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

d, where d is the diameter.

• prism: 2B + Ph, where B is the area of a base, P is the perimeter of a base, and h is the height of the prism.
• pyramid: Failed to parse (Missing texvc executable; please see math/README to configure.): B + \frac{PL}{2}

, where B is the area of the base, P is the perimeter of the base, and L is the length of the slant.

• rectangular prism: Failed to parse (Missing texvc executable; please see math/README to configure.): 2 (\ell w + \ell h + w h)

, where Failed to parse (Missing texvc executable; please see math/README to configure.): \ell

is the length, w is the width, and h is the height.

General formula[link]

The general formula for the surface area of the graph of a continuously differentiable function Failed to parse (Missing texvc executable; please see math/README to configure.): z=f(x,y),

where Failed to parse (Missing texvc executable; please see math/README to configure.): (x,y)\in D\subset\mathbb{R}^2
and Failed to parse (Missing texvc executable; please see math/README to configure.): D
is a region in the xy-plane with the smooth boundary:

Failed to parse (Missing texvc executable; please see math/README to configure.): A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\,dx\,dy.

Even more general formula for the area of the graph of a parametric surface in the vector form Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}=\mathbf{r}(u,v),

where Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{r}
is a continuously differentiable vector function of Failed to parse (Missing texvc executable; please see math/README to configure.): (u,v)\in D\subset\mathbb{R}^2

Failed to parse (Missing texvc executable; please see math/README to configure.): A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv.

^[4]

Minimization[link]

Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles.

The question of the filling area of the Riemannian circle remains open.

See also[link]

• Equi-areal mapping
• Integral
• Orders of magnitude (area)—A list of areas by size.
• Perimeter
• Planimeter, an instrument for measuring small areas, e.g. on maps.
• Volume

References[link]

Notes[link]

External links[link]

Wikimedia Commons has media related to: Area

Look up area in Wiktionary, the free dictionary.

• Weisstein, Eric W., "Area" from MathWorld.
• Area Calculator
• Area formulas
• Conversion cable diameter to circle cross-sectional area and vice versa

This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2010)

Morse code, a famous type of code

A code is a rule for converting a piece of information (for example, a letter, word, phrase, or gesture) into another form or representation (one sign into another sign), not necessarily of the same type.

In communications and information processing, encoding is the process by which information from a source is converted into symbols to be communicated. Decoding is the reverse process, converting these code symbols back into information understandable by a receiver.

One reason for coding is to enable communication in places where ordinary spoken or written language is difficult or impossible. For example, semaphore, where the configuration of flags held by a signaller or the arms of a semaphore tower encodes parts of the message, typically individual letters and numbers. Another person standing a great distance away can interpret the flags and reproduce the words sent.

Theory[link]

It has been suggested that this article or section be merged into Coding theory . (Discuss) Proposed since July 2010.

In information theory and computer science, a code is usually considered as an algorithm which uniquely represents symbols from some source alphabet, by encoded strings, which may be in some other target alphabet. An extension of the code for representing sequences of symbols over the source alphabet is obtained by concatenating the encoded strings.

Before giving a mathematically precise definition, we give a brief example. The mapping

Failed to parse (Missing texvc executable; please see math/README to configure.): C = \{\, a\mapsto 0, b\mapsto 01, c\mapsto 011\,\}

is a code, whose source alphabet is the set Failed to parse (Missing texvc executable; please see math/README to configure.): \{a,b,c\}

and whose target alphabet is the set Failed to parse (Missing texvc executable; please see math/README to configure.): \{0,1\}

. Using the extension of the code, the encoded string 0011001011 can be grouped into codewords as 0 – 011 – 0 – 01 – 011, and these in turn can be decoded to the sequence of source symbols acabc.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively. A code Failed to parse (Missing texvc executable; please see math/README to configure.): C:\, S \to T^*

is a total function mapping each symbol from S to a sequence of symbols over T, and the extension of M to a homomorphism of Failed to parse (Missing texvc executable; please see math/README to configure.): S^*
into Failed to parse (Missing texvc executable; please see math/README to configure.): T^*

, which naturally maps each sequence of source symbols to a sequence of target symbols, is referred to as its extension.

Variable-length codes[link]

In this section we consider codes, which encode each source (clear text) character by a code word from some dictionary, and concatenation of such code words give us an encoded string. Variable-length codes are especially useful when clear text characters have different probabilities; see also entropy encoding.

A prefix code is a code with the "prefix property": there is no valid code word in the system that is a prefix (start) of any other valid code word in the set. Huffman coding is the most known algorithm for deriving prefix codes, so prefix codes are also widely referred to as "Huffman codes", even when the code was not produced by a Huffman algorithm. Other examples of prefix codes are country calling codes, the country and publisher parts of ISBNs, and the Secondary Synchronization Codes used in the UMTS W-CDMA 3G Wireless Standard.

Kraft's inequality characterizes the sets of code word lengths that are possible in a prefix code. Virtually, any uniquely decodable one-to-many code, not necessary a prefix one, must satisfy Kraft's inequality.

Block codes[link]

This section requires expansion.

Error correcting codes[link]

Codes may also be used to represent data in a way more resistant to errors in transmission or storage. Such a "code" is called an error-correcting code, and works by including carefully crafted redundancy with the stored (or transmitted) data. Examples include Hamming codes, Reed–Solomon, Reed–Muller, Walsh–Hadamard, Bose–Chaudhuri–Hochquenghem, Turbo, Golay, Goppa, low-density parity-check codes, and space–time codes. Error detecting codes can be optimised to detect burst errors, or random errors.

Examples[link]

Codes in communication used for brevity[link]

A cable code replaces words (e.g., ship or invoice) with shorter words, allowing the same information to be sent with fewer characters, more quickly, and most important, less expensively.

Codes can be used for brevity. When telegraph messages were the state of the art in rapid long distance communication, elaborate systems of commercial codes that encoded complete phrases into single words (commonly five-letter groups) were developed, so that telegraphers became conversant with such "words" as BYOXO ("Are you trying to weasel out of our deal?"), LIOUY ("Why do you not answer my question?"), BMULD ("You're a skunk!"), or AYYLU ("Not clearly coded, repeat more clearly."). Code words were chosen for various reasons: length, pronounceability, etc. Meanings were chosen to fit perceived needs: commercial negotiations, military terms for military codes, diplomatic terms for diplomatic codes, any and all of the preceding for espionage codes. Codebooks and codebook publishers proliferated, including one run as a front for the American Black Chamber run by Herbert Yardley between the First and Second World Wars. The purpose of most of these codes was to save on cable costs. The use of data coding for data compression predates the computer era; an early example is the telegraph Morse code where more-frequently used characters have shorter representations. Techniques such as Huffman coding are now used by computer-based algorithms to compress large data files into a more compact form for storage or transmission.

Character encodings[link]

Probably the most widely known data communications code so far (aka character representation) in use today is ASCII. In one or another (somewhat compatible) version, it is used by nearly all personal computers, terminals, printers, and other communication equipment. It represents 128 characters with seven-bit binary numbers—that is, as a string of seven 1s and 0s. In ASCII a lowercase "a" is always 1100001, an uppercase "A" always 1000001, and so on. There are many other encodings, which represent each character by a byte (usually referred as code pages), integer code point (Unicode) or a byte sequence (UTF-8).

Genetic code[link]

Biological organisms contain genetic material that is used to control their function and development. This is DNA which contains units named genes that can produce proteins through a code (genetic code) in which a series of triplets {codons} of four possible nucleotides are translated into one of twenty possible amino acids. A sequence of codons results in a corresponding sequence of amino acids that form a protein.

Gödel code[link]

In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem. Here, the idea was to map mathematical notation to a natural number (using a Gödel numbering).

Other[link]

There are codes using colors, like traffic lights, the color code employed to mark the nominal value of the electrical resistors or that of the trashcans devoted to specific types of garbage (paper, glass, biological, etc.)

In marketing, coupon codes can be used for a financial discount or rebate when purchasing a product from an internet retailer.

In military environments, specific sounds with the cornet are used for different uses: to mark some moments of the day, to command the infantry in the battlefield, etc.

Communication systems for sensory impairments, as the sign language for deaf people and braille for blind people, are based in movement or tactile codes.

Musical scores are the most common way to encode music.

Specific games, as chess, have their own code systems to record the matches (chess notation).

Cryptography[link]

In the history of cryptography, codes were once common for ensuring the confidentiality of communications, although ciphers are now used instead. See code (cryptography).

Secret codes intended to obscure the real messages, ranging from serious (mainly espionage in military, diplomatic, business, etc.) to trivial (romance, games) can be any kind of imaginative encoding: flowers, game cards, clothes, fans, hats, melodies, birds, etc., in which the sole requisite is the previous agreement of the meaning by both the sender and the receiver.

Other examples[link]

Other examples of encoding include:

• Encoding (in cognition) is a basic perceptual process of interpreting incoming stimuli; technically speaking, it is a complex, multi-stage process of converting relatively objective sensory input (e.g., light, sound) into subjectively meaningful experience.
• A content format is a specific encoding format for converting a specific type of data to information.
• Text encoding uses a markup language to tag the structure and other features of a text to facilitate processing by computers. (See also Text Encoding Initiative.)
• Semantics encoding of formal language A in formal language B is a method of representing all terms (e.g. programs or descriptions) of language A using language B.
• Electronic encoding transforms a signal into a code optimized for transmission or storage, generally done with a codec.
• Neural encoding is the way in which information is represented in neurons.
• Memory encoding is the process of converting sensations into memories.
• Television encoding: NTSC, PAL and SECAM

Other examples of decoding include:

• Digital-to-analog converter, the use of analog circuit for decoding operations
• Decoding (Computer Science)
• Decoding methods, methods in communication theory for decoding codewords sent over a noisy channel
• Digital signal processing, the study of signals in a digital representation and the processing methods of these signals
• Word decoding, the use of phonics to decipher print patterns and translate them into the sounds of language

Codes and acronyms[link]

Acronyms and abbreviations can be considered codes, and in a sense all languages and writing systems are codes for human thought.

International Air Transport Association airport codes are three-letter codes used to designate airports and used for bag tags. Station codes are similary used on railways, but are usually national, so the same code can be used for different stations if they are in different countries.

Occasionally a code word achieves an independent existence (and meaning) while the original equivalent phrase is forgotten or at least no longer has the precise meaning attributed to the code word. For example, '30' was widely used in journalism to mean "end of story", and it is sometimes used in other contexts to signify "the end".

References[link]

See also[link]

Wikimedia Commons has media related to: Codes

• Asemic writing
• Equipment codes
• Semiotics
• Quantum error correction