# Heptagon

## Definition

A heptagon is a polygon with seven sides and seven angles. The term ‘heptagon’ is derived from the Greek words ‘hepta’ meaning seven and ‘gon’ meaning sides. A heptagon is also known as 7-gon or septagon (‘septa’ means seven in Latin).

The below-given properties and formulas to find the perimeter and area only apply to the regular heptagon.

## Properties

1. Has 7 sides of equal length; in heptagon ABCDEFG, AB = BC = CD = DE = EF = FG = GA
2. Has 7 interior angles each measuring 128.57°; so ∠ABC = ∠BCD = ∠CDE = ∠DEF = ∠EFG =∠FGA = ∠GAB
3. The sum of all seven interior angles is 900°; so ∠ABC + ∠BCD + ∠CDE + ∠DEF + ∠EFG +∠FGH + ∠GAB= 900°
4. Has 7 exterior angles, each measuring 51.43°
5. Has 14 diagonals; shown as AC, AD, AE, AF, BD, BE, BF, BG, CE, CF, CG, DF, DG, and EG

## Formulas

### Perimeter

The formula for finding the perimeter of a heptagon is given below:

Find the perimeter of a regular heptagon having each side measuring 20 cm.

Solution:

As we know,
Perimeter (P) = 7a, here a = 22 cm
= 7 x 22 cm
= 154 cm

### Area

The formula for finding the area of a heptagon is given below:

The above equation is approximately equal to

A = 3.634a2 square units, here cot π/7 = cot 25.71 = 2.0765

Find the area of a regular heptagon having each side measuring 7 cm.

Solution:

As we know,
Area (A) = 7/4 (a2cot π/7)
= 3.634 a2, here a = 7 cm
= 3.634 (7)2
= 3.634 (49)
= 178.06 cm2

### Angles

#### Interior Angle

The angle formed inside the heptagon at its corners when the line segments join in an end to end fashion.

Sum of Interior Angles

The total measure of all the interior angles combined in the heptagon. The formula is given below:

Sum of the interior angles = (n-2) x 180°, here n = number of sides

In heptagon ABCDEFG, n = 7

Thus,

Sum of the interior angles = (7 -2) x 180°

= 900°

One Interior Angle

The measure of one interior angle can be obtained by dividing the sum of the interior angles by the number of sides in a heptagon. The formula is given below:

One interior angle = (n-2) x 180°/n, here n = number of sides

In heptagon ABCDEFG, n = 7

Thus,

Sum of the interior angles = (7-2) x 180°/7

= 128.57°

#### Exterior Angle

The angle formed by any side of the heptagon and the extension of its adjacent side. The formula is given below:

Exterior angle = 360°/n, here n = number of sides

In heptagon ABCDEFG, n = 7

Thus,

Exterior angle = 360°/7

= 51.43°

## Types

Depending on the sides, angles, and vertices, heptagon shapes are classified into the following types:

1. Regular Heptagon: Has seven sides of equal length and seven interior angles each measuring 128.571° and exterior angles of 51.43° each. The sides of a regular heptagon meet each other at an angle of 5π/7 radians or [128(4/7) degrees]. It has seven lines of symmetry and rotational equilibrium of order seven.  All regular heptagons are convex.
2. Irregular Heptagon: Does not have all sides equal or all interior angles equal, but the sum of all seven interior angles is equal to 900°. An irregular heptagon can be both convex and concave.
3. Convex Heptagon: Have all vertices pointing outwards. No interior angle of a convex heptagon measure more than 180°, and all the diagonals lie inside the closed figure. A convex heptagon can be both regular and irregular.
4. Concave Heptagon: Have at least one vertex pointing inwards with an interior angle greater than 180°. At least one diagonal lies outside the closed figure. Thus all concave heptagons are irregular.

## Examples in Real Life

• The United Kingdom currently has two heptagonal-shaped coins, the 50p, and 20p pieces, and the Barbados dollar is also heptagonal. The 1000 Kwacha coin of Zambia is a true heptagon.
• The Brazilian 25-cent coin has a heptagon inscribed in the coin’s disk.
• In architecture, heptagonal floor plans are found in the Mausoleum of Prince Ernst in Stadthagen, Germany.
• Some other manmade objects such as a heptagonal-shaped prism, window, clock, mirror, and lawn.
• Some police batches in the US have a {7/2} heptagram outline.
• The shape of some cactus plants.
• The K7 complete graph is often drawn as a regular heptagon having twenty-one connected edges.