Pentagon


We study several different shapes in Geometry. Polygon is a two-dimensional (2D) shape composed of interior angles and straight lines. Examples of polygons include triangle (a three-sided polygon), quadrilateral (a four-sided polygon), pentagon (a five-sided polygon), hexagon (a six-sided polygon), heptagon (a seven-sided polygon), octagon (an eight-sided polygon), and so on. 

In this article, we will discuss the five-sided polygon, that is, pentagon, along with its shape, properties, types, area, and perimeter.

Definition of Pentagon

A pentagon is one of the many types of polygon having five sides and five angles. The word 'Pentagon' arises from two words, that are, ‘Penta’ and ‘Gonia,’ meaning five angles. In a pentagon, all the sides meet with each other, that too, from end to end to form its shape. Hence, the number of pentagon sides is equal to five. 

The Shape of Pentagon

Like the other polygons, including triangles and quadrilaterals such as rectangle, square, to name a few, the pentagon is also a polygon with five sides and five angles. Depending on the angles, sides, and vertices of a pentagon, there are different types of polygon shapes, namely:

  1. Regular Pentagon and Irregular Pentagon

  2. Convex Pentagon and Concave Pentagon

Regular and Irregular Pentagon

If a pentagon has all five sides of equal lengths, and all the five angles of equal measures, then it is a regular pentagon. If in a pentagon, the sides aren't of equal lengths and angles aren't of equal measures, then it is an irregular pentagon.

Concave and Convex Pentagon

A concave pentagon is the one that has at least one vertex pointing inwards. A convex pentagon is the one that has all its vertices pointing outwards.

Properties of a Pentagon

A pentagon has quite a few interesting properties, which are as follows:

1. In a pentagon, the sum of the measures of the interior angles is equal to 540 degrees.

2. If all the sides of a pentagon are equal in their lengths and all its angles are equal in their measures, then it a regular pentagon. Otherwise, it is an irregular pentagon.

3. The measure of the interior angle of a regular pentagon is 108 degrees, and the measure of the exterior angle of a regular pentagon is 72 degrees.

4. In an equilateral pentagon, there are five sides of equal lengths. 

5. In a rectangular pentagon, the sum of the measures of the interior angles is 540 degrees.

What is the Area of a Pentagon?

The formula to find the area of a regular pentagon with a given side and apothem length (the line from the centre of the pentagon to a side, which intersects the side at a right angle, i.e., at 90 degrees) is as follows:

Area of Pentagon = A = (5/2) * Length of the Side * Apothem Sq Units

The area of a regular pentagon can also be calculated by dividing it into five equal triangles. 

We already know that the area of a triangle is given by the formula – ½ * Base * Height. So, we can find the area of a regular pentagon as – 5 * Area of triangle.

What is the Perimeter of a Pentagon?

As all the sides of a regular pentagon are equal in length, the formula for the perimeter or circumference of a pentagon is as follows:

Perimeter = P = 5a Units, where ‘a’ is the length of the side of the pentagon.

SOLVED EXAMPLES

Question 1

A regular pentagon has a side of length 5cm and length of apothem as 6cm. Find its area and perimeter. 

Answer 1

The length of the side of the pentagon = a = 5cm.

Length of apothem = 6cm.

The formula for finding the area of a pentagon is as follows:

Area of Pentagon = A = (5/2) * Length of the Side * Apothem Sq Units

Substitute the values of the length of the side of the pentagon and the length of apothem in the formula mentioned above.

Area of Pentagon = A = (5/2) * 5 * 6 cm2

Area of Pentagon = A = (5 * 5 * 3) cm2

Area of Pentagon = A = 75 cm2.

The formula for finding the perimeter of a regular pentagon is as follows:

Perimeter = P = 5a Units, where ‘a’ is the length of the side of the pentagon.

Perimeter = P = 5 * 5 cm, where a = length of the side of the pentagon, which is 5cm in this particular case.

Perimeter = P = 25 cm

Hence, the area of the pentagon is 75 cm2, and its Perimeter is 25 cm. 

Question 2

A regular pentagon has a side of length 4cm and length of apothem as 8cm. Find its area and perimeter. 

Answer 2

The length of the side of the pentagon = a = 4cm.

Length of apothem = 8cm.

The formula for finding the area of a pentagon is as follows:

Area of Pentagon = A = (5/2) * Length of the Side * Apothem Sq Units

Substitute the values of the length of the side of the pentagon and the length of apothem in the formula mentioned above.

Area of Pentagon = A = (5/2) * 4 * 8 cm2

Area of Pentagon = A = (5 * 4 * 4) cm2

Area of Pentagon = A = 80 cm2.

The formula for finding the perimeter of a regular pentagon is as follows:

Perimeter = P = 5a Units, where ‘a’ is the length of the side of the pentagon.

Perimeter = P = 5 * 4 cm, where a = length of the side of the pentagon, which is 4cm in this particular case.

Perimeter = P = 20 cm

Hence, the area of the pentagon is 80 cm2, and its Perimeter is 20 cm.