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}.

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

In terms of R (circumradius), the area is

In terms of r (inradius), the area is

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

The area can also be derived as follows:

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.

Thomas Jacob "Jack" Black (born August 28, 1969) is an American comedian, actor and musician. He makes up one half of the comedy and satirical rock duo Tenacious D. The group has three albums as well as a television series and a film. His acting career is extensive, starring primarily as bumbling, cocky, but internally self-conscious outsiders in comedy films. He was a member of the Frat Pack, a group of comedians who have appeared together in several Hollywood films, and has been nominated for a Golden Globe award. He has also won an MTV Movie Award, and a Nickelodeon Kids Choice Award as well as hosting the 2006, 2008, and 2011 Kids Choice Awards.

Black was born in Santa Monica, California, the son of two satellite engineers, Judith (née Cohen), who worked on the Hubble Space Telescope, and Thomas William Black. His mother is Jewish and his father converted to Judaism (Black attended Hebrew school). On The Tonight Show, he said his surname may have come from the occupational name Blacksmith. Black's parents divorced when he was 10; Black moved to Culver City with his father and frequently visited his mother’s home. As a child, Black appeared in a commercial for the Activision game Pitfall! in 1982, which was later shown during an appearance on Jimmy Kimmel Live and again on The Tonight Show with Conan O' Brien. When Black entered high school, his parents enrolled him at the Poseidon School, a private secondary school engineered specifically for students struggling in the traditional school system. Black also attended the Crossroads School, where he excelled in drama. He later attended UCLA but dropped out during his sophomore year to pursue a career in entertainment, at which point his father discontinued his financial support.^{[citation needed]} Fellow UCLA alumnus Tim Robbins later cast Black in Bob Roberts. He also had recurring roles on the HBO sketch comedy series Mr. Show.

Carlos Ray "Chuck" Norris (born March 10, 1940) is an American martial artist and actor. After serving in the United States Air Force, he began his rise to fame as a martial artist and has since founded his own school, Chun Kuk Do. As a result of his "tough guy" image, an Internet phenomenon began in 2005 known as Chuck Norris facts, ascribing various implausible or even impossible feats to Norris.

Norris appeared in a number of action films, such as Way of the Dragon in which he starred alongside Bruce Lee, and was The Cannon Group's leading star in the 1980s. He next played the starring role in the television series Walker, Texas Ranger from 1993 to 2001.

Norris is a devout Christian and politically conservative. He has written several books on Christianity and donated to a number of Republican candidates and causes. In 2007 and 2008, he campaigned for former Arkansas Governor Mike Huckabee, who was running for the Republican nomination for President in 2008. In the 2012 election he endorsed former Speaker of the House Newt Gingrich. Norris also writes a column for the conservative website WorldNetDaily.

Fuerza Guerrera is a Mexican Luchador enmascarado (or masked professional wrestler) who made his debut in 1978. In his career Guerrera has worked for ever major wrestling organization in Mexico including Consejo Mundial de Lucha Libre (CMLL), Asistencia Asesoría y Administración (AAA), Universal Wrestling Association (UWA), World Wrestling Association (WWA) and International Wrestling Revolution Group (IWRG). Throughout his career Fuerza Guerrera has held many titles for CMLL, AAA and IWRG. Fuerza Guerrera's real name is not a matter of public record, as is often the case with masked wrestlers in Mexico where their private lives are kept a secret from the wrestling fans. Only his paternal surname is known, as his son Eduardo Aníbal González Hernández' name is a matter of public record making Fuerza Guerrera's paternal surname "González" as well. Eduardo González is better known as Luchador Juventud Guerrera. Fuerza Guerrera is Spanish for "Warrior's Strength", not to be mistaken for the surname Guerrero.

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. 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).

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.)