104 LearnersLast updated on August 30, 2025
Surface Area of Pentagon
A pentagon is a 2-dimensional polygon with five sides. The surface area of a pentagon refers to the total area covered by its surface. In this article, we will learn about the surface area of a pentagon.
What is the Surface Area of a Pentagon?
The surface area of a pentagon is the total area occupied by the boundary or surface of a pentagon. It is measured in square units.
A pentagon is a 2D shape with five straight sides and five angles.
There are different types of pentagons based on their angles and sides, such as regular pentagons, where all sides and angles are equal, and irregular pentagons, where sides and angles are not equal.
Surface Area of a Pentagon Formula
A pentagon can have different formulas for calculating its area depending on whether it is a regular or irregular pentagon. In this section, we will explore the formula for a regular pentagon. For a regular pentagon with side length \(s\): Area of a regular pentagon = 1/4 √(5(5 + 2√5) s2)
Area of a Regular Pentagon
The area of a regular pentagon can be calculated using the side length of the pentagon. The formula to find the area of a regular pentagon is: Area =1/4 √(5(5 + 2√5) s2)
Where s is the length of a side of the pentagon.
Area of an Irregular Pentagon
The area of an irregular pentagon is more complex to calculate because it does not have equal sides and angles.
One approach is to divide the pentagon into simpler shapes like triangles and rectangles, calculate the area of each, and then sum them up.
Another method would be using the coordinates of the vertices and applying the shoelace formula.
Volume of a Pentagon-Based Prism
The volume of a pentagon-based prism shows how much space is inside it. It tells us how much space the prism can hold.
The volume of a pentagon-based prism can be found by using the formula:
Volume = Base Area × Height
Where the base area is the area of the pentagon base, which can be calculated using the formula for regular or irregular pentagons as described above.
Confusion between Regular and Irregular Pentagons
Students assume all pentagons are regular, but irregular pentagons have unequal sides and angles. Always verify if the pentagon is regular before using specific formulas.
Mistake 1
Using Incorrect Formula for Side Length
Some students mistakenly use the wrong formulas for the side length when calculating the area of a pentagon. Remember to use the correct formula based on whether the pentagon is regular or irregular.
Mistake 2
Neglecting Units of Measurement
A common mistake of students is ignoring the units of measurement. Always ensure that all measurements are in the same unit when calculating area to avoid errors.
Mistake 3
Miscalculating the Square Root Term
Students often miscalculate the square root term in the area formula for a regular pentagon. Double-check your calculations and ensure that you handle the square root and other operations correctly.
Mistake 4
Assuming All Pentagon Areas Are the Same
Some students mistakenly believe that the area of any pentagon is calculated the same way. Regular and irregular pentagons have different methods for calculating area.
Mistake 5
Solved Examples of Surface Area of Pentagon
Find the area of a regular pentagon with a side length of 8 cm.
Area = 110.11 cm²
Problem 1
Given \(s = 8\) cm. Use the formula: Area = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 8^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 64\) = 110.11 cm²
Calculate the area of a regular pentagon with a side length of 10 cm.
Explanation
Area = 172.05 cm²
Problem 2
Use the formula: Area = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 10^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 100\) = 172.05 cm²
A regular pentagon has a perimeter of 30 cm. Find its area.
Explanation
Area = 61.94 cm²
Problem 3
First, find the side length: Perimeter = 5s 30 = 5s s = 6 cm Then, use the area formula: Area = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 6^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 36\) = 61.94 cm²
Find the area of a regular pentagon with side length 12 cm.
Explanation
Area = 248.89 cm²
Problem 4
Area = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 12^2\) = \(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times 144\) = 248.89 cm²
The area of a regular pentagon is 150 cm². Find the side length.
Explanation
Side length = 9.13 cm
It is the total area that covers the surface of the pentagon, measured in square units.
1.What are the two types of pentagons?
2.How is the area of a regular pentagon calculated?
3.How do you find the area of an irregular pentagon?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Pentagon
Students often make mistakes while calculating the surface area of a pentagon, which leads to wrong answers. Below are some common mistakes and the ways to avoid them.
Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
