Heptagon

Heptagon is a two-dimensional shape with seven angles, seven vertices, and seven edges. This seven-sided polygon “heptagon” is made up of two words ‘Hepta’ and ‘Gonia’, which means seven angles. Another name given to it is septagon or 7-gon. A heptagon has fourteen diagonals. A polygon is a closed two-dimensional shape made up of straight sides having any number of sides. In simple words, we can say that a heptagon is a polygon with 7 sides.

In this article, we will explore the properties and shape of the heptagon. We will discuss about its sides, interior angles, diagonals, and vertices. We will solve a few examples related to the concept for a better understanding.

A heptagon is a seven-sided polygon that has seven angles, seven vertices, and seven edges. They may have the same or different dimensions of length. It is a closed figure and a heptagon with all equal seven sides is called a regular heptagon. Let’s observe the figure given below that shows a heptagon.

Heptagon Sides

The seven sides of a heptagon are straight edges and can be of the same or different lengths. These sides meet each other but do not intersect or cross each other. The heptagon sides meet at the vertices to form a seven-sided closed figure.

Heptagon Angles

A heptagon has seven interior angles and the sum of all interior angles is equal to 900°. Some angles of a the figure can be obtuse or acute. The sum of the exterior angles of a heptagon is equal to 360° and this holds for both regular and irregular heptagons.

Heptagon Diagonals

A heptagon has fourteen diagonals. For a convex heptagon, the diagonals lie inside the figure whereas for a concave heptagon, at least one diagonal lies outside the figure.

Heptagon shapes can be categorized based on their sides and angles.

I) Based on the side lengths, heptagons can be classified as follows:

• Regular Heptagon: A regular heptagon is one that has equal sides and equal angles. The sum of all the interior angles of a polygon is equal to (n - 2) × 180°, where n is the number of sides. Since a heptagon has 7 sides, the sum of its interior angles is equal to (7 - 2) × 180° = 5 × 180° = 900°. The value of each interior angle of a regular heptagon is 900°/7 = 128.57°

• Irregular Heptagon: An irregular heptagon is one that has sides and angles of different measures. The value of each interior angle of an irregular heptagon will be different. However, the sum of all the interior angles of an irregular heptagon is also 900°.

The following figures show a regular and an irregular heptagon.

II) Based on angle measures, heptagons can be classified as follows:

• Convex Heptagon: A convex heptagon has all the interior angles measure less than 180°. They can either be regular or irregular heptagons. All the vertices of the convex heptagon are pointed outwards.

• Concave Heptagon: In a concave heptagon at least one of the interior angles is greater than 180°. They can either be regular or irregular heptagons. At least one vertex points inwards in a concave heptagon.

The following figures show a convex and a concave heptagon.

Now that we know the basic meaning of a heptagon, let us now look into some important properties of a heptagon as follows:

• A heptagon has 7 sides, 7 edges, and 7 vertices.
• The sum of the interior angles of a heptagon is equal to 900°.
• The value of each interior angle of a regular heptagon is equal to 128.57°
• The sum of exterior angles of a heptagon is equal to 360°
• The number of diagonals that can be drawn in a heptagon is 14.
• The measure of the central angle of a regular heptagon is approximately equal to 51.43 degrees.
• A regular heptagon is also known as a convex heptagon since all its interior angles are less than 180°
• An irregular heptagon has unequal sides and angles of different measures.

There are many formulas related to a regular heptagon. Let us understand how to find the perimeter and area of a regular heptagon using the heptagon formulas.

Perimeter of a Heptagon

We know that a regular heptagon has 7 sides of equal length. Therefore, the perimeter of a regular heptagon is given as 7 × Side length. Therefore, the perimeter of a regular heptagon with side length ‘a’ is given as, Perimeter = 7a

Area of a Heptagon

Area of a heptagon is defined as the total space occupied by the polygon. The area of a regular heptagon with side length ‘a’ is calculated using the formula, Area = (7a²/4) cot (π/7). This formula can be simplified and approximately written as 3.634a², where 'a' is the side length. We can use this to calculate the area of a regular heptagon.

A heptagon consists of 7 interior angles and 7 exterior angles. Let's read about the interior and exterior angles of a heptagon.

Interior Angles of a Regular Heptagon

The sum of interior angles of a regular polygon is given using the interior angle formula that is (n - 2) × 180º where n is the number of sides of the polygon. Thus, for a heptagon, n = 7. Sum of interior angles of a regular heptagon = (7 - 2) × 180º = 900º. Thus, each interior angle of a regular heptagon = 900/7 = 128.57º

Exterior Angles of a Regular Heptagon

According to the sum of exterior angles formula, the sum of all the exterior angles of a regular polygon is equal to 360º. Thus, the sum of all the exterior angles of a regular heptagon is equal to 360º. Thus, each exterior angle of a regular heptagon = 360/7 = 51.43º

Important Notes on Heptagon

• A heptagon has 7 edges, 7 interior angles, and 7 vertices.
• The sum of interior angles of the heptagon is 900° and the sum of its exterior angles is 360°.
• A heptagon can be of two types - regular and irregular on the basis of its sides.

Related Articles

• Polygons
• Hexagon
• Interior Angles

What is a Heptagon in Math?

A heptagon is a seven-sided polygon that has 7 sides, 7 vertices, and 7 edges. The side lengths and angle measures of a heptagon may or may not be equal to each other. Another name given to heptagon is septagon.

What is a Convex Heptagon?

A convex heptagon is one in which all the interior angles are less than 180°. All the vertices of a convex heptagon point towards the outward direction.

How to Calculate Heptagon Diagonals?

The diagonals of a heptagon can be calculated by drawing line segments connecting every two opposite vertices and counting them. We can also calculate the number of diagonals by using the formula n(n - 3)/2 where n is the number of sides of a polygon. For a heptagon, the value of n is 7. Thus, the number of diagonals can be calculated as 7(7 - 3)/2 = 28/2 = 14.

How many Diagonals does a Heptagon have?

A heptagon has 14 diagonals in all which can be calculated using the formula, n(n - 3)/2 where n is the number of sides.

What is the Sum of the Angle Measures in a Heptagon?

The sum of the interior angles of a heptagon is equal to 900° which can be calculated using the interior angle formula of a regular polygon, (n - 2) × 180° where n is the number of sides. For a heptagon, the value of n is 7. Thus, by using the formula, the sum of the interior angles will be 900°. The sum of the exterior angles of a heptagon is 360°.

How to Find the Interior Angles of a Regular Heptagon?

Each interior angle of a regular heptagon can be calculated using the sum of interior angles of a heptagon which is 900°. Since all the angles of a regular heptagon are equal, and there are 7 interior angles in a heptagon, the value of each interior angle will be 900°/7 = 128.57°

How to Find the Perimeter of a Regular Heptagon?

Perimeter is the sum of all the sides of a polygon. Since a regular heptagon has 7 sides of equal length, its perimeter is calculated using the formula, perimeter of regular heptagon = 7 × side length.

How to Categorize Heptagons on the Basis of Heptagon Sides?

The heptagons can be categorized as Regular and Irregular Heptagon on the basis of their side lengths. If all seven sides have equal length, then it is called a regular heptagon. On the other hand, if at least one side of the heptagon has a different length, then it is an irregular heptagon.

What are the Types of Heptagon?

We can categorize heptagon on the basis of sides and angles. We have mainly four types of heptagons:

• Regular Heptagon
• Irregular Heptagon
• Concave Heptagon
• Convex Heptagon