Heptagon

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

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

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 ${\displaystyle \pi /7,}$ 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 ${\displaystyle {\tfrac {7R^{2}}{2}}\sin {\tfrac {2\pi }{7}},}$ while the area of the circle itself is ${\displaystyle \pi R^{2};}$ thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.

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 ${\displaystyle \scriptstyle {2\cos {\tfrac {2\pi }{7}}\approx 1.247}}$ 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.

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

An animation from a neusis construction with radius of circumcircle ${\displaystyle {\overline {OA}}=6}$, according to Andrew M. Gleason[1] based on the angle trisection by means of the tomahawk. This construction relies on the fact that ${\displaystyle 6\cos \left({\frac {2\pi }{7}}\right)=2{\sqrt {7}}\cos \left({\frac {1}{3}}\arctan \left(3{\sqrt {3}}\right)\right)-1.}$

Heptagon with given side length:
An animation from a neusis construction with marked ruler, according to David Johnson Leisk (Crockett Johnson).

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.[2][3] Let A lie on the circumference of the circumcircle. Draw arc BOC. Then ${\displaystyle \scriptstyle {BD={1 \over 2}BC}}$ gives an approximation for the edge of the heptagon.

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

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

Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.[4]

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.[5]

a=red, b=blue, c=green lines

The regular heptagon's side a, shorter diagonal b, and longer diagonal c, with a<b<c, satisfy[6]^{: Lemma 1}

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

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

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

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

and hence

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

and[6]^{: Coro. 2}

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

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

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

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

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

We also have[7]

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

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

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

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 ${\displaystyle \pi /7,2\pi /7,}$ and ${\displaystyle 4\pi /7.}$ Thus its sides coincide with one side and two particular diagonals of the regular heptagon.[6]

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