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Regular heptagon calculator

- By Dr. Minas E. Lemonis, PhD - Updated: April 24, 2020

This tool calculates the basic geometric properties of a regular heptagon. Regular polygons are equilateral (all sides equal) and their angles are equal too. The tool can calculate the properties of the heptagon, given either the length of its sides, or the inradius or the circumradius or the area or the height or the width. Enter below the shape dimensions. The calculated results will have the same units as your input. Please use consistent units for any input.

Known data:
Geometric properties:
Area =
Perimeter =
α =
Ri =
Rc =

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Bounding box:
Height h =
Width w =
Angles :
Interior φ =
Central θ =
shape details
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Table of Contents
See also
Properties of Pentagon
Properties of Hexagon
Properties of Octagon
Properties of Decagon
Properties of a N-gon
All Geometric Shapes

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Theoretical background

Table of contents

Definitions

Heptagon is a polygon with seven sides and seven vertices. Like any polygon, a heptagon may be either convex or concave, as illustrated in the next figure. When it is convex, all its interior angles are lower than 180°. On the other hand, when its is concave, one or more of its interior angles is larger than 180°. When all the edges of the heptagon are equal then it is called equilateral. An equilateral heptagon can be either convex or concave. When all the edges are equal and additionally all interior angles are equal then the heptagon is a regular one. A regular heptagon is convex by default. The next figure illustrates the classification of heptagons, also presenting some concave equilateral ones that are non regular. Any heptagon that is not regular is called irregular

Heptagon types

The sum of the internal angles of a heptagon is constant and equal to 900°. This is a common characteristic of every heptagon, regular or irregular, convex or concave. The truth of this property can be easily discovered if we divide the heptagon to individual, non overlapping triangles. To do this, we have to draw straight lines between all the vertices, avoiding any intersections. At the end, the heptagon is divided into five triangles, as shown in the figure below. Taking into account that in a single triangle the internal angles sum up to is 180°, it is concluded that for 5 triangles the internal angles should sum up to 5x180°=900° or 5\pi .

Properties or regular heptagons

Symmetry

A regular heptagon features seven axes of symmetry. Every axis passes through a vertex of the heptagon and the midpoint of the opposite edge, as presented in the following drawing. All axes of symmetry intersect at a common point, the center of the regular heptagon. which is also the center of gravity or centroid of the shape.

Interior and central angle

By definition the interior angles of a regular heptagon are equal. It is also a common property of all heptagons that the sum of their interior angles is always 900° (or 5\pi ), as explained previously. Therefore, the interior angle, \varphi , of a regular heptagon should be:

\varphi={900^\circ\over7}={5\pi\over7}

which is approximated to:

\varphi=128.57^\circ .

If we draw straight lines from the center of the regular heptagon, towards every vertex, seven identical, isosceles, triangles are defined. The angle of each triangle, opposite from the heptagon edge, is called central angle theta of the regular heptagon. Since, there are seven central angles around the center and they are all equal, each one of them should be equal to:

\theta={360^\circ\over7}={2\pi\over7}

which is approximated to:

\theta\approx51.43^\circ

It is not coincidence that the interior and central angle sum up to \pi :

\varphi+\theta={5\pi\over7}+{2\pi\over7}=\pi .

In other words \phi and \theta are supplementary angles.

Circumcircle and incircle

For every regular polygon, a circle can be drawn that touches all the polygon vertices. This is the, so called, cirmuscribed circle of the regular polygon and is also a characteristic property of the regular heptagon too. Commonly, the circumscribed circle is also referred to a circumcircle. The center of this circle is also the center of the heptagon, where all the symmetry axes are intersecting. The radius of circumcircle, R_c , is usually called circumradius.

Another characteristic circle of regular polygons, and consequently regular heptagon too, is the, so called inscribedcircle or incircle, in short. The incircle is tangent to all edges touching them at their midpoint. Its radius, R_i , is usually called inradius. The incircle and the circumcircle share the same center. 

The following drawing illustrates both the circumcircle and the incircle of the regular heptagon.

Circumscribed and inscribed circle of regular heptagon

The circumradius R_c and inradius R_i are related to the length of the regular polygon edge a . In this section we will try to establish these relationships for the regular heptagon. The highlighted triangle, in the following figure will be examined. One vertex of the triangle is actually the heptagon center. As a result, one of its edges, that lands to a heptagon vertex, should be equal to the circumradius R_c and the other one, that lands to the middle of the heptagon edge, should be equal the inradius R_i . The latter is also a bisector of central angle \theta , and therefore, the interior angle of the triangle, by the center, should be \theta/2 . Also, the triangle is a right one, since by definition the incircle is tangential to the edges of the polygon at their midpoint.

Using basic trigonometry we find:

\begin{split} R_c & = \frac{a}{2 \sin{\frac{\theta}{2}}} \\ R_i & = \frac{a}{2 \tan{\frac{\theta}{2}}} \\ R_i & = R_c \cos{\frac{\theta}{2}} \end{split}

where \theta the central angle and a the side length. It turns out that these expressions are valid for any regular polygon, not just the heptagon. We can obtain specific expression for the regular heptagon by setting \theta=2\pi/7 . These specific formulas for the regular heptagon are:

\begin{split} R_c & = \frac{a}{2 \sin(\pi/7)} \approx 1.152 a \\ R_i & = \frac{a}{2 \tan(\pi/7)} \approx 1.038 a \\ \\ R_i & = R_c \cos(\pi/7) \approx 0.901 R_c \end{split}

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Area and perimeter

The area of the regular heptagon (or any regular polygon) can be expressed in terms of the edge length a . This can be achieved, if we divide the shape in simpler subareas. In fact, the regular heptagon is divided to seven identical isosceles triangles, if we draw straight lines from the center, towards every vertex. These lines are radii of the circumcircle and therefore, have length R_c . The heights of the triangles (from the heptagon center towards the opposite edge), are also medians and indeed, radii of the incircle, with length equal to R_i (since the incircle is tangential to all sides of the heptagon touching them at their midpoints). The area of each triangle is then:

A_1=\frac{1}{2}a R_i .

Therefore, the total area of the seven triangles is found:

A =7A_1=7\frac{1}{2} a R_i \Rightarrow

A= {7\over2}a\frac{a}{2 \tan(\pi/7)}\Rightarrow

A=\frac{7a^2}{4 \tan(\pi/7)}

An approximation of the last relationship is:

A \approx 3.634 a^2

The perimeter of any N-sided regular polygon is simply the sum of the lengths of all edges: P = N a . Therefore, for the regular heptagon, width seven edges :

P = 7a

Bounding box

The bounding box of a planar shape is the smallest rectangle that encloses the shape completely. For the regular heptagon the bounding box may be drawn intuitively, as shown in the next figure. Its dimensions, namely the height h and the width w , depend on the heptagon edge length a . These relationships are examined next.

Height

The height h of the regular heptagon is the distance from one of its vertices to the opposite edge. It is indeed perpendicular to the opposite edge and passes through the center of the heptagon. By definition though, the distance from the center to a vertex is the circumradius R_c of the heptagon while the distance from the center to an edge is the inradius R_i . Therefore the following expression is derived:

h=R_c+R_i

It is possible to express the height h in terms of the circumradius R_c , or the inradius R_i or the side length a , using the respective analytical expressions for these quantities. The following formulas are derived:

h=R_c\left(1+\cos(\theta/2)\right)

h=R_i\left(1+{1\over \cos(\theta/2)}\right)

h={a\over2}{1+\cos(\theta/2)\over\sin(\theta/2)}

where \theta=2\pi/7 .

Substituting the value of \theta , of the regular heptagon, to the last expressions we get the following approximations:

h\approx 1.901 R_c

h\approx 2.110 R_i

h\approx 2.191 a

Width

The width w is the distance between two opposite vertices of the regular heptagon (the length of its diagonal). In order to find this distance we will employ the right triangle highlighted with dashed line, in the figure above.

The hypotenuse of the triangle is actually the side length of the heptagon, which is a . Also, one of the triangle angles is supplementary to the adjacent interior angle \varphi of the heptagon. It has been explained before, though, that the supplementary of \varphi is indeed the central angle \theta . Therefore, we may find the length w_1 of the triangle side:

w_1=a \cos\theta

Finally, we can determine the total width w by adding twice the length w_1 to the side length a (due to symmetry the triangle to the right of the heptagon is identical to the one examined):

w=a+2a \cos\theta

Substituting, \theta=2π/7 we get an approximation of the last formula:

w=2.247a

Examples

Example 1

Determine the circumradius, the inradius and the area of a regular heptagon, with side length a=10''

We will use the exact analytical expressions for the circumradius and the inradius, in terms of the side length a , that have been described in the previous sections. These are:

R_c = \frac{a}{2 \sin(\pi/7)}

R_i = \frac{a}{2 \tan(\pi/7)}

Since the side length a is given, all we have to do is substitute its value 10'' to these expressions. If your calculator expects degrees for trigonometric functions, the angle \pi/7 is approximately 25.71°. The circumradius is:

R_c= \frac{10''}{2 \sin(\pi/7)}\approx 11.52'' ,

and the inradius:

R_i= \frac{10''}{2 \tan(\pi/7)}\approx 10.38'' .

The area of a regular heptagon is also given in terms of the side length a , by this formula:

A = \frac{7a^2}{4 \tan(\pi/7)}

Substituting a=10'' we find:

A = \frac{7\ (10'')^2}{4 \tan(\pi/7)} \approx 363.4\ \text{in}^2

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Example 2

What is the edge length a of the following regular heptagons:

  1. having area A=120\ \textrm{in}^2
  2. having height h=16''
  3. having width w=10''
1. Regular heptagon with given area

The area of a regular heptagon A , in terms of the side length a , is given by the equation:

A = \frac{7a^2}{4 \tan(\pi/7)}

Since we are looking for a , we have to rearrange the formula:

a^2 = \frac{4A}{7} \tan(\pi/7) \Rightarrow

a =+\sqrt{ \frac{4A}{7} \tan(\pi/7) }

From the last equation we can calculate the required side length a , if we substitute A=120\ \textrm{in}^2 :

a =+\sqrt{ \frac{4\times 120\ \textrm{in}^2}{7} \tan(\pi/7) }\approx5.746\ \textrm{in}

2. Regular heptagon with given height

The height of the regular heptagon is related to the side length a with the equation:

h=\frac{a\left(1+\cos(\pi/7)\right)}{2\sin(\pi/7)}

Rearranging:

a={2h\sin(\pi/7) \over 1+\cos(\pi/7)}

From the last expression we can calculate the required side length a , if we substitute h=16'' :

a={2\times16''\sin(\pi/7) \over 1+\cos(\pi/7))}\approx7.304''

3. Regular heptagon with given width

The width of the regular heptagon is related to the side length a with the formula:

w=a+2a \cos(2\pi/7)

Therefore:

a=\frac{w}{1+2\cos(2\pi/7)}

From the last equation we can calculate the required side length a , if we substitute w=10'' :

a=\frac{10''}{1+2\cos(2\pi/7)}\approx 4.450''

Regular heptagon cheat-sheet

In the following table a concise list of the main formulas, related to the regular heptagon is included. Also some approximations that may prove handy for practical problems are listed too.

Regular heptagon quick reference

Circumradius: R_c=\frac{a}{2 \sin(\pi/7)}
Inradius: R_i=\frac{a}{2 \tan(\pi/7)}
Height: h=R_c+R_i
Width: w=a+2a \cos(2\pi/7)
Area: A=\frac{7a^2}{4 \tan(\pi/7)}
Interior angle: \varphi=5\pi/7
Central angle: \theta=2\pi/7
Approximations:

R_c \approx 1.152a

R_i \approx 1.038a

h \approx 2.191a

w \approx 2.247a

A\approx 3.634 a^2

\theta=51.43^\circ

\varphi=128.57^\circ

{\pi\over7}=25.71^\circ

See also

Properties of Pentagon
Properties of Hexagon
Properties of Octagon
Properties of Decagon
Properties of a N-gon
All Geometric Shapes

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See also
Properties of Pentagon
Properties of Hexagon
Properties of Octagon
Properties of Decagon
Properties of a N-gon
All Geometric Shapes