heptagon n : a seven-sided polygon

English

Etymology

Since 16th century, from ἑπτάγωνον (heptagonon), from ἑπτά (hepta), seven, + γωνία (gonia), angle.

Noun

1. A polygon with seven sides and seven angles.

Translations

Related terms

• heptagonal
• septagon

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, 128.5714286 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.

The heptagon is also sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix). The OED lists "septagon" as meaning "heptagonal".

Construction

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. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge 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.

Heptagrams

Two kinds of heptagrams can be constructed from regular heptagons, labeled by Schläfli symbols , and , with the divisor being the interval of connection.

• Heptagonal tiling

External links

• Definition and properties of a heptagon With interactive animation
• Approximate construction method
• Another approximate construction method
• Polygons - Heptagons