Heptagon

Types of Polygons

In Geometry, we study about different types of shapes. A polygon is any closed geometrical shape made up of straight lines that can be drawn on a flat surface, like a piece of paper. A circle or any other shape that includes a curve are not polygons.

Polygons are classified into different types depending on the number of sides they have: (image will be updated soon)

  1. Triangle (Polygon having 3 sides and 3 interior angles)

  2. Quadrilateral (Polygon having 4 sides and 4 interior angles)

  3. Pentagon (Polygon having 5 sides and 5 interior angles)

  4. Hexagon (Polygon having 6 sides and 6 interior angles)

  5. Heptagon (Polygon having 7 sides and 7 interior angles)

  6. Octagon (Polygon having 8 sides and 8 interior angles)

  7. Nonagon (Polygon having 9 sides and 9 interior angles)

  8. Decagon (Polygon having 10 sides and 10 interior angles)


In General, A Polygon with ‘n’ Sides Has: 

  1. ‘n’ interior angles. 

  2. Sum of interior angles = (n - 2) × 180°

  3. Each interior angle of a regular polygon = \[\frac{(n-2) х 180⁰}{n}\]

  4. Sum of exterior angles = 360°


In this article, we will learn about the seven-sided polygon called “heptagon” with its proper definition, shape, number of sides, properties, its formula of perimeter and area in detail.


Definition of Heptagon

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A heptagon is a polygon which has seven sides and seven angles. The word “heptagon” is made up of two words, namely ‘Hepta’ and ‘Gonia’, which means seven angles. 

Sometimes the heptagon is also known as “septagon”.

Since, heptagon has 7 sides therefore,

  1. Sum of interior angles = (n - 2) × 180°

                                     = (7 - 2) × 180° = 5 × 180°

                                 = 900°

  1. Each interior angle of a regular heptagon = \[\frac{(n-2) х 180⁰}{n}\]

                                                                 = \[\frac{(7-2) х 180⁰}{5}\] = \[\frac{900}{5}\]

                                                                 = 128.571°

  1. Sum of exterior angles = 360°


Shapes of Heptagon

Depending on the sides, angles and vertices, heptagon shapes are classified as:

  1. Regular heptagons

  2. Irregular heptagons


Regular Heptagon

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To be a regular heptagon the heptagon must have:

  1. seven congruent sides (sides of equal length)

  2. seven congruent interior angles (each measuring 128.571°)

  3. seven congruent exterior angles of 51.428°

Note: Regular heptagons do not have parallel sides.


Irregular Heptagon

Irregular heptagons are the heptagon having different side lengths and angle measures. (image will be updated soon)

Irregular heptagons can be a convex heptagon or a concave heptagon:

  • Convex Heptagon – A heptagon having not any internal angles more than 180°.

  • Concave Heptagon – A heptagon having one interior angle more than 180°. (image will be updated soon)


Properties of Heptagon

  1. It has seven sides, seven vertices and seven interior angles.

  2. It has 14 diagonals.

  3. The sum of all interior angles is 900°.

  4. The sum of the exterior angles is 360°.

  5. Regular heptagon has all seven sides of equal length.

  6. Each interior angle of a regular heptagon measures 128.571°.

  7. Irregular heptagons have different side lengths and angle measures.

  8. All diagonals of the convex heptagon lie inside the heptagon.

  9. some diagonals of concave heptagon may lie outside the heptagon. 


The Perimeter of a Heptagon

The perimeter of a heptagon is the sum of lengths of its seven sides.

For a regular heptagon, since the length of all seven sides are equal. 

Therefore, the perimeter of a regular heptagon = 7 × (side length) units.  


Area of a Heptagon

Area of the heptagon is the region covered by the sides of the heptagon.

For a regular heptagon, its area can be calculated by:

(image will be updated soon)

  1. If Measure of Side Length and Apothem are Given, Then:

(Apothem: a line from the centre of a regular polygon at right angles to any of its sides.)

Area of heptagon = \[\frac{7}{2}\] × (side length) × (apothem) units2

OR, 

Area of heptagon = \[\frac{1}{2}\]× (perimeter of heptagon) × (apothem) units2

  1. If Only Measure of Side Length is Given, Then:

Area of heptagon = \[\frac{7}{4}\] cot\[\frac{π}{7}\]⁰ × (side length) 2 units2

Where, cot\[\frac{π}{7}\]⁰  = cot 25.714⁰ = 2.0765

OR,

Area of heptagon = 3.634 × (side length) 2 units2


Solved Problems:

Q.1. Find the perimeter and area of a regular heptagon whose side is 7 cm?

Solution: Given, side of heptagon = 7 cm 

Perimeter of a regular heptagon = 7 × (side length) units

                                                        = 7 × 7 cm

                                                        = 49 cm

And,

Area of heptagon = 3.634 × (side length) 2 units2

                                                = 3.634 × (7)2

                               = 178.066 cm2

Hence, the perimeter and area of a regular heptagon whose side is 7 cm is 40 cm and 140 cm2 respectively.