The concept of the area comes under the topic of mensuration which is the branch of geometry that deals with the measurements of shapes i.e., length, area, perimeter, volume etc. for any two-dimensional (2D) or three-dimensional (3D) figures. In this article, we are going to discuss the area of the Pentagon which is a Geometrical 2D figure with 5 sides.
Area of a Pentagon
The Pentagon area is the region that is enclosed within the 5 sides of a Pentagon.
The area of the pentagon is calculated by using two approaches. Those are-
- Using the length of one side.
- Using Side and Apothem length.
Area of Pentagon using side length
This approach is used when we had the length of a side of a pentagon. To find the area of the pentagon using the side length the formula is given by,
Area of Pentagon = (1/4) * sqrt(5 * (5 + 2 * sqrt(5))) * (Side)^2
Area of Pentagon using Side and Apothem length
This approach is used to calculate the area of the pentagon when the side and apothem length of a pentagon is known. The formula of this approach is easy when compared to the above approach formula. It is given by-
Area of Pentagon = (5/2) × (side length) × (Apothem length)
To get more grip on this concept let's look at a few examples.
Sample Problems
Problem 1: What is the area of the pentagon with a side of length 5 cm.
Solution:
Given,
Side length (s)= 5cm
Area of Pentagon = (1/4) (√(5(5+2√5))) s2
= (1/4) (√(5(5+2√5))) (5)2
The value for the expression (1/4) (√(5(5+2√5))) is approximately equal to 1.72
= 1.72 × 25
= 43 cm2
The area of pentagon with side 5 cm is 43 cm2.
Problem 2: What is the area of the pentagon with a side of length 6.5 cm.
Solution:
Given,
Side length (s)= 6.5cm
Area of Pentagon = (1/4) (√(5(5+2√5))) s2
= (1/4) (√(5(5+2√5))) (6.5)2
The value for the expression (1/4) (√(5(5+2√5))) is approximately equal to 1.72
= 1.72 × 42.25
= 72.67cm2
The area of pentagon with side 5cm is 72.67cm2.
Problem 3: Find the area of the Pentagon whose length of side and apothem are 5cm,3cm respectively.
Solution:
Given
Side length (s)= 5cm
Apothem length (a)=3cm
Area of pentagon = (5/2) x s x a
= (5/2) x 5 x 3
= 75/2
= 42.5 cm2.
The area of given pentagon is 42.5cm2.
Problem 4: Find the area of the Pentagon whose length of the side is 6cm and apothem length is 5cm?
Solution:
Given
Side length (s)= 6cm
Apothem length (a)=5cm
Area of pentagon = (5/2) x s x a
= (5/2) x 6 x 5
= 150/2
= 75 cm2.
The area of given pentagon is 75cm2.
Problem 5: What is the area of the Pentagon whose length of the side is 4cm and length of apothem is 2 cm.
Solution:
Given
Side length (s)= 4cm
Apothem length (a)=2cm
Area of pentagon = (5/2) x s x a
= (5/2) x 4 x 2
= 20 cm2.
The area of given pentagon with side 4cm and apothem 2 cm is 20 cm2.
Unsolved Practice Problems
- Calculate the area of a pentagon with a side length of 7 cm.
- Find the area of a pentagon with a side length of 8 cm.
- Determine the area of a pentagon whose side length is 9 cm and apothem length is 4 cm.
- What is the area of a pentagon with a side length of 10 cm and an apothem length of 6 cm?
- Calculate the area of a pentagon with side length 3 cm and apothem length 1.5 cm.
Summary
In this article, we discussed the concept of the area of a pentagon. We explored two main methods to calculate the area: using the length of one side and using the side and apothem length. Sample problems with detailed solutions were provided to illustrate each approach.