Regular octagon
A regular octagon
Type	Regular polygon
Edges and vertices	8
Schläfli symbol	{8} t{4}
Coxeter–Dynkin diagram
Symmetry group	D8, order 2×8
Internal angle (degrees)	135°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is a polygon that has eight sides. A regular octagon is represented by the Schläfli symbol {8}.

Regular octagon[link]

A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles is 1080° (as for any octagon). The area of a regular octagon of side length a is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427125\,a^2.

In terms of R (circumradius), the area is

Failed to parse (Missing texvc executable; please see math/README to configure.): A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \simeq 2.828427\,R^2.

In terms of r (inradius), the area is

Failed to parse (Missing texvc executable; please see math/README to configure.): A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \simeq 3.3137085\,r^2.

These last two coefficients bracket the value of pi, the area of the unit circle.

The area of a regular octagon can be computed as a truncated square.

The area can also be derived as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!A=S^2-a^2,

where S is the span of the octagon, or the second shortest diagonal; and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45–45–90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.

Given the length of a side a, the span S is:

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

Failed to parse (Missing texvc executable; please see math/README to configure.): S \approx 2.414a

The area is then as above:

Failed to parse (Missing texvc executable; please see math/README to configure.): A=((1+\sqrt{2})a)^2-a^2=2(1+\sqrt{2})a^2

Failed to parse (Missing texvc executable; please see math/README to configure.): A \approx 4.828a^2

Expressed in terms of the span, area is:

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

Failed to parse (Missing texvc executable; please see math/README to configure.): A \approx 0.828S^2

Another simple formula for the area is

Failed to parse (Missing texvc executable; please see math/README to configure.): \ A=2ad

where d is the distance between parallel sides (the same as span S in the diagram).

Construction[link]

A regular octagon is constructible with compass and straightedge.

Standard coordinates[link]

The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:

• (±1, ±(1+√2))
• (±(1+√2), ±1).

Uses of octagons[link]

• 

  In many parts of the world, stop signs are in the shape of a regular octagon.

• 

  Umbrellas often have an octagonal outline.

• 

  The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.

• 

  The street & block layout of Barcelona's Eixample district is based on non-regular octagons

• 

  Vichy Pastilles, octagon-shaped candies.

• 

  Janggi uses octagonal pieces.

• 

  Japanese lottery machines often have octagonal shape.

Derived figures[link]

• 

  An eight-sided star, called an octagram, with Schläfli symbol {8/3} is contained with a regular octagon.

• 

  The vertex figure of the uniform polyhedron, great dirhombicosidodecahedron is contained within an irregular 8-sided star polygon, with four edges going through its center.

• 

  An octagonal prism contains two octagons.

• 

  The truncated square tiling has 2 octagons around every vertex.

• 

  The truncated cuboctahedron contains 6 octagons.

• 

  An octagonal antiprism contains two octagons.

Petrie polygons[link]

The octagon is the Petrie polygon for these 12 higher dimensional uniform polytopes, shown in these skew orthogonal projections of in A₇, B₄, and D₅ Coxeter planes.

A7	7-simplex	Rectified 7-simplex	Birectified 7-simplex	Trirectified 7-simplex
B4	16-cell	Rectified 16-cell	Rectified tesseract	Tesseract
D5	Trirectified demipenteract	Birectified demipenteract	Rectified demipenteract	Demipenteract

See also[link]

• Octagram
• Octagonal number
• Octagon house
• Oktogon, a major intersection in Budapest, Hungary
• Bumper pool
• Rub el Hizb (also known as Al Quds Star and as Octa Star)
• Smoothed octagon

References[link]

External links[link]

• Octagon Calculator
• Definition and properties of an octagon With interactive animation
• Weisstein, Eric W., "Octagon" from MathWorld.

The black king has been checkmated; the game is over.

Checkmate (frequently shortened to mate) is a situation in chess (and in other boardgames of the chaturanga family) in which one player's king is threatened with capture (in check) and there is no way to meet that threat. Or, simply put, the king is under direct attack and cannot avoid being captured. Delivering checkmate is the ultimate goal in chess: a player who is checkmated loses the game. In normal chess the king is never actually captured – the game ends as soon as the king is checkmated because checkmate leaves the defensive player with no legal moves.^{[note 1]} In practice, most players resign an inevitably lost game before being checkmated. It is considered bad etiquette to continue playing in a completely hopeless position (Burgess 2000:481).^{[note 2]}

If a king is under attack but the threat can be met, then the king is said to be in check, but is not in checkmate. If a player is not in check but has no legal move (that is, every possible move would put the king in check), the result of the game is stalemate, and the game ends in a draw. (See rules of chess.)

A checkmating move is denoted in algebraic chess notation with the hash symbol (#) – for example, 34.Qh8# or by "++". (The symbol "++" is sometimes used to indicate double check.)

This article uses algebraic notation to describe chess moves.

Examples[link]

Fool's mate

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

White has been checkmated

D. Byrne vs. Fischer

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

After 41... Rc2, checkmate

Checkmate with a rook

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Black has been checkmated

A checkmate may occur in as few as two moves with all of the pieces still on the board (as in Fool's mate, in the opening phase of the game), in a middlegame position (as in the 1956 game called the Game of the Century between Donald Byrne and Bobby Fischer),^[1] or after many moves with as few as three pieces in an endgame position.

History[link]

In early Sanskrit chess (ca. 500-700) the king could be captured and this ended the game. The Persians (ca. 700-800) introduced the idea of warning that the king was under attack (announcing check in modern terminology). This was done to avoid the early and accidental end of a game. Later the Persians added the additional rule that a king could not be moved into check or left in check. As a result, the king could not be captured (Davidson 1949:22). Checkmate was thus the logical and only decisive way of ending a game (since if it was checkmate, any move would be illegal) (Davidson 1949:63–64).

Before about 1600 the game could also be won by capturing all of the opponent's pieces other than the king (annihilation or robado) (see bare king). In Medieval times players began to consider it nobler to win by checkmate, so annihilation became a half-win for a while, until it was abandoned (Davidson 1949:63–64).

Etymology[link]

The term checkmate is an alteration or Hobson-Jobson of the Persian phrase "Shāh Māt" which means, literally, "the King is helpless" (or "ambushed", "defeated", or "stumped", but not "dead").^[2][3] It is a common misconception that it means "the King is dead", as chess reached Europe via the Islamic world, and Arabic māta مَاتَ means "died" or "is dead" (Hooper & Whyld 1992), (Davidson 1949:70), (Sunnucks 1970), (McKean 2005), (Golombek 1976:27).^[4]

Moghadam traced the etymology of the word mate. It comes from a Persian verb mandan, meaning "to remain", which is cognate with the Latin word manco. It means "remained" in the sense of "abandoned" and the formal translation is "surprised", in the military sense of "ambushed" (not in the sense of "astonished"). So the king is in mate when he is ambushed, at a loss, or abandoned to his fate (Davidson 1949:70–71).

The term "checkmate" also has origins in the term "check", which means temporarily stop, and which also is in the chessplaying context related to the term "chess", which derives from the French word "echecs", which is derived from the alternating black and white pattern of an eschequier, or counting table, a pattern which was duplicated in larger form in some countries on the floor of the exchequer, a central bank, and whence such related terms as "checkers" and "checkered" were brought about.

The term checkmate in modern parlance is a dead metaphor for an irrefutable and strategic victory.^[5]

Two major pieces[link]

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

White checkmates easily

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Checkmate with queen and rook

Two major pieces (queens or rooks) can easily force checkmate on the edge of the board, even without the help of their king. The process is to put the two pieces on adjacent ranks or files and gradually force the king to the side of the board, where one piece keeps the king on the edge of the board while the other delivers checkmate (Pandolfini 1988:18–20).

In the first diagram, White checkmates easily by forcing the black king to the edge a rank at a time or a file at a time:

1. Qg5+ Kd4

2. Rf4+ Ke3

3. Qg3+ Ke2

4. Rf2+ Ke1

5. Qg1# (second diagram) (Silman 2007:7–8).

The checkmate with two queens or with two rooks is similar (Pandolfini 1988:20).

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Mid-board checkmate with king and two rooks

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Mid-board checkmate with two queens

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Another mid-board checkmate with two queens

Checkmate can be forced even away from the edge of the board with two rooks and a king, or with a queen, rook, and king, while two queens are able to force checkmate in the center without the help of the king (Pandolfini 2009:32–33).

Basic checkmates[link]

Here are the common fundamental checkmates when one side has only his king and the other side has only the minimum material needed to force checkmate, i.e. (1) one queen, (2) one rook, (3) two bishops on opposite-colored squares, or (4) a bishop and a knight. The king must help in accomplishing all of these checkmates. If the superior side has more material, checkmates are easier (Silman 2007:33).

The checkmate with the queen is the most important, but it is also very easy to achieve. It often occurs after a pawn has queened. The next most important one is the checkmate with the rook, and it is also very easy to achieve. The checkmates with the two bishops and with a bishop and knight are not nearly as important, since they only occur infrequently. The two bishop checkmate is fairly easy to accomplish, but the bishop and knight checkmate is difficult and requires precision.

King and queen[link]

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Checkmate with the queen

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

A second checkmate with the queen

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

A third type of checkmate

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

A fourth type of checkmate

The first two diagrams show representatives of the basic checkmate positions with a queen, which can occur on any edge of the board. Naturally, the exact position can vary from the diagram. In the first of the checkmate positions, the queen is directly in front of the opposing king and the white king is protecting its queen. In the second checkmate position, the kings are in opposition and the queen mates on the rank (or file) of the king. See Wikibooks - Chess/The Endgame for a demonstration of how the king and queen versus king mate is achieved.

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

White checkmates easily

With the side with the queen to move, checkmate can be forced in at most ten moves from any starting position, with optimal play by both sides, but usually fewer moves are required (Fine & Benko 2003:1–2). (Müller & Lamprecht 2001:16). In positions in which a pawn has just promoted to a queen, at most nine moves are required (Levy & Newborn 1991:144). In this position, White checkmates by confining the black king to a rectangle and shrinking the rectangle to force the king to the edge of the board:

1. Qf6 Kd5

2. Qe7 Kd4

3. Kc2 Kd5

4. Kc3 Kc6

5. Kc4 Kb6

6. Qd7 Ka6

7. Qb5+ Ka7

8. Kc5 Ka8

9. Kc6 Ka7

10. Qb7# (Seirawan 2003:4–5).

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Stalemate if Black is to move. The white king can be anywhere (except a7, b7, or b8, which would be an illegal position).

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Stalemate if Black is to move

The superior side must be careful to not stalemate the opposing king, whereas the defender would like to get into such a position. There are two general types of stalemate positions that can occur, which the winning side must avoid (Fine & Benko 2003:2).

King and rook[link]

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

Checkmate with the rook