Heptagon

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In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5π/7 radians, approximately 128.6 degrees. Its Schläfli symbol is . The area of a regular heptagon of side length a is given by

[A = \fraca^2 \cot \frac \simeq 3.63391 a^2.]

A regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. This type of construction is called a Neusis construction. The impossibility of a ruler and compass construction follows from the observation that 2cos(2π/7) ≈ 1.247 is a zero of the irreducible cubic x³ + x² - 2x - 1. Consequently this polynomial is the minimal polynomial of 2cos(2π/7), whereas the degree of the minimal polynomial for a constructible number must be a power of 2.

The United Kingdom currently (2006) has two heptagonal coins, the 50p and 20p pieces. The 20 eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a curvilinear heptagon to make them curves of constant width: the sides are curved outwards so that the coin will roll smoothly in vending machines.

The heptagon is also sometimes referred to as the septagon, using "sept-" (an elision of , a Latin-derived numerical prefix) for seven, rather than , a Greek-derived numerical prefix.

A Neusis construction of the interior angle in a regular heptagon.

Heptagon

See also