Heptagon

The heptagon is sometimes referred to as the septagon, using septa- (an elision of septua-), a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix (both are cognate), together with the suffix -gon for Greek: γωνἰα, romanized: gonía, meaning angle.

Regular heptagon

A regular heptagon, in which all sides and all angles are equal, has internal angles of ⁠${\displaystyle \frac{5}{7}\pi }$⁠ radians (⁠${\displaystyle 128\frac{4}{7}}$⁠ degrees). Its Schläfli symbol is {7}.

Area

The area (A) of a regular heptagon of side length a is given by:

${\displaystyle A=\frac{7}{4}{a}^2\cot \frac{1}{7}\pi \simeq 3.634a^2.}$

This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side. The apothem is half the cotangent of ⁠${\displaystyle \frac{1}{7}\pi }$⁠, and the area of each of the 14 small triangles is one-fourth of the apothem.

The area of a regular heptagon inscribed in a circle of radius R is ${\displaystyle \frac{7}{2}{R}^2\sin \frac{2}{7}\pi ,}$ while the area of the circle itself is ${\displaystyle \pi R^2;}$ thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.

Construction

As 7 is a Pierpont prime but not a Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. It is the smallest regular polygon with this property. 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 ${\displaystyle 2\cos \frac{2}{7}\pi \approx 1.247}$ is a zero of the irreducible cubic x³ + x² − 2x − 1. Consequently, this polynomial is the minimal polynomial of ⁠${\displaystyle 2\cos \frac{2}{7}\pi }$⁠, whereas the degree of the minimal polynomial for a constructible number must be a power of 2.

Approximation

An approximation for practical use with an error of about 0.2% is to use half the side of an equilateral triangle inscribed in the same circle as the length of the side of a regular heptagon. It is unknown who first found this approximation, but it was mentioned by Heron of Alexandria's Metrica in the 1st century AD, was well known to medieval Islamic mathematicians, and can be found in the work of Albrecht Dürer. Let A lie on the circumference of the circumcircle. Draw arc BOC. Then ${\displaystyle BD=\frac{1}{2}BC}$ gives an approximation for the edge of the heptagon.

This approximation uses ${\displaystyle \frac{1}{2}\sqrt{3}\approx 0.86603}$ for the side of the heptagon inscribed in the unit circle while the exact value is ${\displaystyle 2\sin \frac{1}{7}\pi \approx 0.86777}$.

Example to illustrate the error: At a circumscribed circle radius r = 1 m, the absolute error of the first side would be approximately −1.7 mm

Other approximations

There are other approximations of a heptagon using compass and straightedge, but they are time-consuming to draw.

Symmetry

The regular heptagon belongs to the D_7h point group (Schoenflies notation), order 28. The symmetry elements are: a 7-fold proper rotation axis C₇, a 7-fold improper rotation axis, S₇, 7 vertical mirror planes, σ_v, 7 2-fold rotation axes, C₂, in the plane of the heptagon and a horizontal mirror plane, σ_h, also in the heptagon's plane.

Diagonals and heptagonal triangle

The regular heptagon's side a, shorter diagonal b, and longer diagonal c, with a<b<c, satisfy

${\displaystyle a^2=c(c-b),}$

${\displaystyle b^2=a(c+a),}$

${\displaystyle c^2=b(a+b),}$

${\displaystyle \frac{1}{a}=\frac{1}{b}+\frac{1}{c}}$ (the optic equation)

and hence

${\displaystyle ab+ac=bc,}$

and

${\displaystyle b^3+2b^{2}c-b{c}^2-c^3=0,}$

${\displaystyle c^3-2c^{2}a-c{a}^2+a^3=0,}$

${\displaystyle a^3-2a^{2}b-a{b}^2+b^3=0,}$

Thus –b/c, c/a, and a/b all satisfy the cubic equation ${\displaystyle t^3-2t^2-t+1=0.}$ However, no algebraic expressions with purely real terms exist for the solutions of this equation, because it is an example of casus irreducibilis.

The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by

${\displaystyle b\approx 1.80193\cdot a,c\approx 2.24698\cdot a.}$

We also have

${\displaystyle b^2-a^2=ac,}$

${\displaystyle c^2-b^2=ab,}$

${\displaystyle a^2-c^2=-bc,}$

and

${\displaystyle \frac{b^2}{a^2}+\frac{c^2}{b^2}+\frac{a^2}{c^2}=5.}$

A heptagonal triangle has vertices coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles ⁠${\displaystyle \frac{1}{7}\pi }$⁠, ⁠${\displaystyle \frac{2}{7}\pi }$⁠, and ⁠${\displaystyle \frac{4}{7}\pi }$⁠. Thus its sides coincide with one side and two particular diagonals of the regular heptagon.

In polyhedra

Apart from the heptagonal prism and heptagonal antiprism, no convex polyhedron made entirely out of regular polygons contains a heptagon as a face.