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# Heptagon

Geometry is the study of shapes. It deals with construction of geometrical shapes and problems based on those. Polygons are an important class of shapes that are used quite often in geometry. Etymologically, polygon word is composed of two Greek words - poly and gonia. Poly meaning many and gonia referring to angles. In this way, the meaning of polygon is one that has many angles.

The shapes that are bounded by at least three straight lines and have at least three interior angles are known as polygons.
The examples of most commonly used polygons are - Triangle (a three-sided polygon), square (a four-sided polygon), pentagon (a five-sided polygon), hexagon (a six-sided polygon), heptagon (a seven-sided polygon), octagon (an eight-sided polygon) and so on.

In this page, we are going to understand about heptagons. Heptagon is one of the types of polygon which has 7 sides and seven angles. A heptagon may be a regular or an irregular in shape. A regular heptagon has 7 equal sides and 7 angles of equal measure. An irregular heptagon does not have all the angles and sides equal. It is irregular in shape.

A regular heptagon is shown in the figure given below.

Heptagon :

The measure of each interior angle of a regular heptagon is about 128.57 degrees or approximately 129 degrees. A heptagon is also called as Septagon.

## Heptagon Definition

A Heptagon is a polygon with 7 sides and 7 angles. If heptagon is regular, then all sides and all angles are equal and the sides meet at an angle of $\frac{5\pi}{7}$ radians.

Heptagon Sides = 7

## Properties of a Heptagon

Given below are some heptagon facts:
Interior angle of a polygon is measured with the help of a general formula which is given by $\frac{180n - 360}{n}$. Here, n is the number of sides. For polygon of 7 sides, interior angle is calculated as below.

$\frac{180n - 360}{n}$ = $\frac{180 \times 7 - 360}{7}$

= $\frac{1260 - 360}{7}$ = $\frac{900}{7}$

= 128.571° (Approximately)

When the sides of a polygon is denoted as S, the area of the heptagon is given by 3.633S2 in approximate calculation. To find the number of diagonals in the polygon, the general formula is $\frac{1}{2}$ $n(n - 3)$, where, n is the number of sides. Therefore, the number of diagonals for polygon of 7 sides is calculated as below.
$\frac{1}{2}$ $n(n - 3)$ = $\frac{1}{2}$ $7(7 - 3)$

= $\frac{1}{2}$ $7(4)$ = $\frac{28}{2}$

= 14 Diagonals

To find the number of triangles in a polygon, the general formula is n - 2, where n is the number of sides. Therefore, the number of triangles in a polygon of 7 sides is calculated as below.

n - 2 = 7 - 2 = 5 triangles

The measure of the central angle of a regular heptagon is about 51.43 degrees.
The number of diagonal of the heptagon = 14
An Irregular Heptagon has sides of different lengths.

## Heptagon Construction

The polygon of seven sides is not easily constructed with the help of compass. But, it can be easily done with the help of a marked ruler and a compass. Also, the compass, straightedge and the trisector are used to construct polygon of seven sides. Construction of the polygon by this method is called as Neusis Construction.

## Area of Heptagon

If the side of a heptagon is denoted as a, the area of the regular heptagon is given by the following formula,

### Heptagon Area Formula:

Area of the heptagon is based upon its nature.

Case 1:
If Heptagonal is regular.

Then, the area of heptagonal = $\frac{1}{2}$ P * a

Where, "P" is the Perimeter and "a" is the apothem of the heptagonal.

Case 2:
If Heptagonal is irregular.

Then, the heptagon divided into 7 triangles and finding the area of each triangle and sum up to find the area of heptagonal.

## Heptagon Angles

Hexagonal is a polygon with 7 angles.

### Heptagon Interior Angles:

Sum of interior angles of an n sided polygon = (2n - 4) * 90 degrees. (In general)

Let us find the sum of the interior angles of a heptagon.

Heptagon is a polygon with 7 sides.

=> n = 7

Sum of interior angles of a heptagon = (2 * 7 - 4) * 90 degrees.

= 10 * 90

= 900 degrees.

If heptagon is regular, then the value of an interior angle is $\frac{900}{7}$ degrees = 128.57 degrees.

## Convex Heptagon

A convex heptagon is a polygon with seven sides in which all of its diagonals lies inside the Heptagon.

Regular Heptagons are Convex Heptagons.

## Concave Heptagon

A Concave Heptagon is a polygon with one or more interior angles greater than 180 degrees and some diagonals will lie outside the polygon. Heptagons that are not convex are concave Heptagons.

Some of the heptagon diagonals lies outside the polygon and heptagon vertices are marked in blue color.

## Heptagon Diagonals

The number of diagonals in a regular polygonal = $\frac{n(n - 3)}{2}$, where n = number of sides of the polygonal.

In case of heptagon, n = 7

=> The number of diagonals in a regular heptagon = $\frac{7(7 - 3)}{2}$ = 14.

## Heptagon Tessellation

A tessellation is a way to tile a floor with regular shapes so that there are no gaps. When we try to fit regular heptagons around a point, we see that regular pentagons overlap.

And, we also know the interior angles sum of a heptagon = 900$^o$.

=> Each interior angle = $\frac{900}{7}$ = 128.57 (not a factor of 360$^o$)

$\therefore$ A regular heptagon does not tesselate.

## Heptagon Examples

Given below are some of the examples on heptagon.

### Solved Examples

Question 1: Find the approximate area of a regular heptagon with a radius of 20 m.
Solution:
Radius of the heptagon = 20 m (Given)

And, n = 7 (Polygonal is heptagon)

We know that, Area of regular heptagon = $\frac{1}{2}$ $r^2$ n sin($\frac{2\pi}{n}$).

= $\frac{1}{2}$ $20^2$ * 7 sin($\frac{2\pi}{7}$)

= 1285.24 sq m.

Question 2: Find the perimeter for the heptagon.

Solution:
Perimeter = Sum of all sides of the heptagon = 1 + 8 + 4 + 5 + 6 + 2 + 9 = 35 cm.

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