Table of Contents

Last modified on March 28th, 2023

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.

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

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

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

The above equation is approximately equal to

A = 3.634a^{2} 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 (a^{2}cot π/7)

= 3.634 a^{2}, here a = 7 cm** = **3.634 (7)^{2}

= 3.634 (49)

= 178.06 cm^{2}

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°

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°

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

**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.**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.**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.**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.

- 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.

Last modified on March 28th, 2023

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