Heptagon

is {7}.

Area

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

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

This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with

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

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.

Construction

As 7 is a

. 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

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 ${\displaystyle 6\cos \left({\frac {2\pi }{7}}\right)=2{\sqrt {7}}\cos \left({\frac {1}{3}}\arctan \left(3{\sqrt {3}}\right)\right)-1.}$

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

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

Other approximations

There are other approximations of a heptagon using compass and straightedge, but they are time consuming to draw. ^[4]

Symmetry

The regular heptagon belongs to the

The regular heptagon's side a, shorter diagonal b, and longer diagonal c, with a<b<c, satisfy^{[7]: 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^{[7]: 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^[8]

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

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

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

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.^[7]

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.

Star heptagons

Two kinds of star heptagons (heptagrams) can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the interval of connection.

Blue, {7/2} and green {7/3} star heptagons inside a red heptagon.

Tiling and packing

Triangle, heptagon, and 42-gon vertex

Hyperbolic heptagon tiling

A regular triangle, heptagon, and 42-gon can completely fill a plane vertex. However, there is no tiling of the plane with only these polygons, because there is no way to fit one of them onto the third side of the triangle without leaving a gap or creating an overlap. In the hyperbolic plane, tilings by regular heptagons are possible. There are also concave heptagon tilings possible in the Euclidean plane.^[9]

The regular heptagon has a double lattice packing of the Euclidean plane of packing density approximately 0.89269. This has been conjectured to be the lowest density possible for the optimal double lattice packing density of any convex set, and more generally for the optimal packing density of any convex set.^[10]

Empirical examples

Heptagon divided into triangles, clay tablet from Susa, 2nd millennium BCE

Heptagonal dome of the Mausoleum of Prince Ernst

The United Kingdom, as of 2022^[update], has two heptagonal

The

A number of coins, including the 20 euro cent coin, have heptagonal symmetry in a shape called the Spanish flower.

In architecture, heptagonal floor plans are very rare. A remarkable example is the Mausoleum of Prince Ernst in Stadthagen, Germany.

Many police badges in the US have a {7/2} heptagram outline.

See also

• Heptagram
• Polygon

References

External links

Look up heptagon in Wiktionary, the free dictionary.

• Definition and properties of a heptagon With interactive animation
• Heptagon according Johnson
• Another approximate construction method
• Polygons – Heptagons
• Recently discovered and highly accurate approximation for the construction of a regular heptagon.
• Heptagon, an approximating construction as an animation
• A heptagon with a given side, an approximating construction as an animation

Heptagon