101 LearnersLast updated on August 30, 2025
Surface Area of Pentagonal Pyramid
A pentagonal pyramid is a 3-dimensional shape with a pentagonal base and triangular faces that meet at a common vertex. The surface area of the pentagonal pyramid is the total area covered by its outer surface, including both its triangular faces and its base. In this article, we will learn about the surface area of a pentagonal pyramid.
What is the Surface Area of a Pentagonal Pyramid?
The surface area of a pentagonal pyramid is the total area occupied by the boundary or surface of the pyramid. It is measured in square units. A pentagonal pyramid has a pentagonal base and triangular faces that connect the base to a single vertex.
The pyramid has two surface areas: the lateral surface area, which includes only the triangular faces, and the total surface area, which includes both the base and the lateral surface area.
Surface Area of a Pentagonal Pyramid Formula
A pentagonal pyramid has a lateral surface and a base, resulting in two types of surface areas: the lateral surface area and the total surface area.
Look at the pentagonal pyramid below to see its surface area, height(h), slant height(l), and base side length(s).
A pentagonal pyramid has two types of surface areas: Lateral Surface Area of a Pentagonal Pyramid Total Surface Area of a Pentagonal Pyramid
Lateral Surface Area of a Pentagonal Pyramid
The area of the triangular faces of the pentagonal pyramid, excluding its base, is known as the lateral surface area of a pentagonal pyramid.
The formula for the lateral surface area (LSA) of the pentagonal pyramid is given as: Lateral Surface Area = 1/2 * perimeter of base * slant height
Here, the perimeter of the base is the sum of all the side lengths of the pentagonal base, and the slant height is the height of each triangular face.
Total Surface Area of a Pentagonal Pyramid
The total area occupied by the pentagonal pyramid, including the area of the base and the area of the triangular faces, is known as the total surface area of the pentagonal pyramid.
The total surface area is calculated by using the formula: Total Surface Area = Base Area + Lateral Surface Area
To find the total surface area of a pentagonal pyramid, calculate the area of the pentagonal base and add it to the lateral surface area.
Volume of a Pentagonal Pyramid
The volume of a pentagonal pyramid shows how much space is inside it.
It can be calculated using the formula: Volume = 1/3 * Base Area * Height
Where the Base Area is the area of the pentagonal base, and Height is the perpendicular distance from the base to the vertex.
Confusion between LSA and TSA
Students assume that the lateral surface area (LSA) and the total surface area (TSA) of a pentagonal pyramid are the same.
This confusion arises because both involve the slant height.
Always remember that LSA is used only for the triangular faces, and TSA includes the base and the triangular faces.
Mistake 1
Using edge length instead of slant height
Some students mistakenly use the base edge length instead of the slant height when finding the lateral surface area.
Remember the formula for lateral surface area is 1/2 * perimeter of base * slant height, so always use the slant height.
Mistake 2
Incorrect base area calculation
A common mistake is incorrectly calculating the area of the pentagonal base.
Use the appropriate formula for the area of a regular pentagon or divide it into triangles if necessary for accuracy.
Mistake 3
Ignoring the base area in total surface area
Students often calculate only the lateral surface and forget to add the pentagonal base.
Always include both parts when calculating the total surface area.
Total surface area = lateral surface area + base area.
Mistake 4
Confusing lateral surface area and triangular face area
Some students mistakenly believe that lateral surface area and triangular face area are different, but for the pyramid, all triangular faces together form the lateral surface area.
Mistake 5
Solved Examples of Surface Area of Pentagonal Pyramid
Find the lateral surface area of a pentagonal pyramid with a base side length of 6 cm and a slant height of 10 cm.
LSA = 150 cm²
Problem 1
The perimeter of the base = 5 * side = 5 * 6 = 30 cm. Use the formula: LSA = 1/2 * perimeter of base * slant height = 1/2 * 30 * 10 = 150 cm²
Find the total surface area of a pentagonal pyramid with a base side length of 4 cm, slant height of 9 cm, and base area of 27.5 cm².
Explanation
TSA = 112.5 cm²
Problem 2
Use the formula: TSA = Base Area + LSA Perimeter of base = 5 * 4 = 20 cm LSA = 1/2 * 20 * 9 = 90 cm² TSA = 27.5 + 90 = 117.5 cm²
A pentagonal pyramid has a base side length of 5 cm and a height of 12 cm.
The base area is 43.01 cm². Find the total surface area.
Explanation
TSA = 173.01 cm²
Problem 3
Find the slant height using the Pythagorean theorem: l = √(height² + (base side/2)²) l = √(12² + (5/2)²) l = √(144 + 6.25) l = √150.25 l ≈ 12.25 cm Use the TSA formula: TSA = Base Area + LSA Perimeter of base = 5 * 5 = 25 cm LSA = 1/2 * 25 * 12.25 = 153.125 cm² TSA = 43.01 + 153.125 = 196.135 cm²
Find the lateral surface area of a pentagonal pyramid with a base side length of 3.5 cm and a slant height of 8 cm.
Explanation
LSA = 70 cm²
Problem 4
Perimeter of base = 5 * 3.5 = 17.5 cm LSA = 1/2 * 17.5 * 8 = 70 cm²
The slant height of a pentagonal pyramid is 15 cm, and its lateral surface area is 225 cm². Find the perimeter of the base.
Explanation
Perimeter = 30 cm
It is the total area that covers the outside of the pyramid, including its triangular faces and the pentagonal base.
1.What are the two types of surface area in a pentagonal pyramid?
2.What is the difference between slant height and edge length?
3.Is lateral surface area the same as the area of the triangular faces?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Pentagonal Pyramid
Students often make mistakes while calculating the surface area of a pentagonal pyramid, 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
