In geometry, any polygon with seven sides and seven angles and seven vertices is called as Heptagon. The sum of angles in a Heptagon is 900°.

In real life, Heptagon shape can be seen in British fifty pence coin which is an equilaterally curved heptagon. The dome in the former Melbourne Magistrates Court in Melbourne, Australia, is a good architectural example for a regular heptagon.
A simple tablet organizer is also a good example for regular Heptagon.

## Regular Heptagon

A Heptagon having all sides of equal length and equal angles is called regular Heptagon.

The figure above shows a regular Heptagon. Each of its internal angles is equal to 128.571° (The formula to find an interior angle of an n-sided polygon is given by $\frac{(n-2)\pi}{n}$). In a polygon, the central angle is the angle made at the centre of the polygon by any two adjacent vertices and is given by the formula $\frac{360^o}{n}$. Thus the central angle measure of a regular Heptagon is 51.429°.
The number of distinct diagonals that are possible to draw from all vertices in a polygon is given in general by the formula $\frac{n(n–3)}{2}$. Thus the number of diagonals that can be drawn in a heptagon are 14.

The number of triangles that can be drawn by drawing the diagonals from a given vertex in general is given by n–2 and so we can draw 5 triangles in a regular Heptagon.

## Irregular Heptagon

An Irregular Heptagon has sides of different lengths.

Irregular polygons are not symmetric and so are not considered as having a centre. Thus have no central angle.

## Convex Heptagon

Geometrically, in any polygon if any line segment drawn between any two interior points lies entirely inside the polygon, then it is called a convex polygon. Similarly convex Heptagon is a 2D geometric figure in which any line segment drawn between any two interior points lies inside the Heptagon. Any regular Heptagon is a Convex Heptagon.

In a convex polygon with n sides, the sum of interior angles is given by the following equation: S = (n -2) × 180°.

So, in a convex Heptagon, the sum of angles is given by (7 -2) × 180° = 5 × 180° = 900°.

In a convex Heptagon, the sum of the degree measures of the exterior angles, one at each vertex, is 360°.

## Concave Heptagon

A concave polygon has one or more vertices which point towards the interior and looks like “pushed in”. A concave Heptagon is defined as a polygon with one or more interior angles greater than 180°. Also a Heptagon that is not convex is a concave Heptagon.

In this 3-pointed heptagon, we can see that the interior angles at B, D, E, G in figure 1 and B, C in the figure 2 are greater than 180°. On observing these figures we can see that in a concave polygon some diagonals will lie outside the polygon, like diagonals AC, CF and FA in figure 1 and AD, BD in figure 2.

## Area of a Heptagon Formula

The area of a Heptagon can be found in different ways. We can use various formulas depending on the data when the heptagon is regular.
Case (i) When the side length, s, is given the area of any n sided polygon is given by,
Area = $\frac{s^{2}n}{(4tan (\frac{180}{n}))}$, where the tangent function is always calculated in degrees.
Putting n = 7, we get the area of regular heptagon with side length s as, Area = 3.633 s2.

Case (ii) When an Apothem ‘a’(Apothem of a regular polygon is a perpendicular line segment drawn from the centre to the midpoint of
one of its sides) is given then the area of a polygon is given by :
Area = $a^{2}ntan(\frac{180}{n})$ where n is the number of sides of a polygon.
Putting n = 7, we get the area of regular heptagon with Apothem ‘a’, Area = 3.371 a2.

Case(iii) When a Radius ‘r’(Radius of a regular polygon is a distance from the centre to any of the vertex) is given, then the area of a polygon is given by
Area =$(\frac{1}{2}) r^{2}nSin(\frac{360}{n})$ where n is the number of sides of a polygon.
Putting n = 7, we get the area of regular heptagon with radius r, Area = 2.7364 r2.
If p is the Perimeter and A is the apothem, the area is also given by Area = ½ P A

In case of an irregular Heptagon, area can be found by breaking the heptagon into 7 triangles and finding the area of each triangle and adding them up.

## Interior Angles of a Heptagon

In a polygon, the interior angles always add up to a constant value and depends only on the number of sides no matter if it regular or irregular, convex or concave, or what size and shape it is. Each interior angle of any regular Heptagon is equal to 128.57° (An interior angle of any polygon is given by $\frac{(180n–360)}{n}$. Here n=7 for a heptagon).

## Exterior Angles of a Heptagon

An exterior angle of a Heptagon is formed by extending any one of its sides. The non-straight angle adjacent to an interior angle is the exterior angle. Each exterior angle of any regular Heptagon is 51.43° (In a polygon, an exterior angle forms a linear pair with its interior angle, so in general the exterior angle is given by 180-interior angle). In a convex polygon, the sum of the exterior angles is 360°.

## Heptagon Examples

Now let us learn how to find area of heptagon with following examples,

### Solved Examples

Question 1: Find the area of Heptagon to nearest tenth with a side length of 16 cms.
Solution:

The length of each side of a heptagon, s = 16 cms
Area of Heptagon = $\frac{s^{2}n}{4tan(\frac{180}{n})}$ = $\frac{s^{2}7}{4tan(\frac{180}{7})}$ =3.6339 s2
= 3.6339 X 16 X 16
= 930.28
= 930.28 sq.cms.
Thus the area of a heptagon with side length as 16 cms is 930.3 sq.cms.

Question 2: Find the area of a regular Heptagon with a radius of 10cms to the nearest hundredth.
Solution:

The radius of a heptagon,r = 10cms
Area of heptagon  =$(\frac{1}{2}) r^{2}nSin(\frac{360}{n})$ = $(\frac{1}{2}) r^{2}7Sin(\frac{360}{7})$  = 2.7364 r2
= 2.7364  x (10)2.
= 273.64 sq.cm
Thus the area of a heptagon with radius of 10 cms is 273.64 sq.cm.

### Octagon

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