109 LearnersLast updated on August 5th, 2025
Surface Area of Prisms and Pyramids
Prisms and pyramids are 3-dimensional shapes that have flat polygonal bases. The surface area of these solids is the total area covered by their outer surfaces, which includes the lateral surfaces and the bases. In this article, we will learn about the surface area of prisms and pyramids.
What is the Surface Area of Prisms and Pyramids?
The surface area of a prism or a pyramid is the total area occupied by the boundary or surface of the shape. It is measured in square units.
A prism is a 3D shape with two identical polygonal bases connected by rectangular lateral faces, while a pyramid has one polygonal base and triangular lateral faces meeting at a common vertex.
Prisms and pyramids are classified based on the shape of their base, such as rectangular, triangular, or hexagonal.
Surface Area of Prisms and Pyramids Formula
Both prisms and pyramids have lateral surfaces and bases contributing to their total surface area. Look at the examples below to see their surface areas, heights, slant heights (for pyramids), and base dimensions.
Prisms have: -
- Lateral Surface Area
- Total Surface Area
Pyramids have: -
- Lateral Surface Area
- Total Surface Area
Lateral Surface Area of Prisms and Pyramids
The lateral surface area of a prism is the sum of the areas of its rectangular faces, while for a pyramid, it is the sum of the areas of its triangular faces excluding the base.
The formula for the lateral surface area depends on the shape of the base and the height or slant height. For a right prism with perimeter P of the base and height h:
Lateral Surface Area = P × h
For a regular pyramid with perimeter P of the base and slant height l:
Lateral Surface Area = 1/2 × P × l
Total Surface Area of Prisms and Pyramids
The total surface area of a prism or pyramid includes both the lateral surface area and the area of the base(s).
The total surface area can be calculated by adding the lateral surface area to the base area.
For a prism: Total Surface Area = Lateral Surface Area + 2 × Base Area
For a pyramid: Total Surface Area = Lateral Surface Area + Base Area
Volume of Prisms and Pyramids
The volume of a prism or pyramid shows how much space is inside it. It tells us the capacity of the shape.
The volume of a prism is the base area multiplied by the height, while the volume of a pyramid is one-third of the base area multiplied by the height.
For a prism: Volume = Base Area × Height
For a pyramid: Volume = 1/3 × Base Area × Height
Confusion between Lateral Surface Area and Total Surface Area
Students assume that the lateral surface area and the total surface area are the same. This confusion arises because both involve the base perimeter or dimensions. Always remember that lateral surface area includes only the sides, while total surface area includes both the sides and the base(s).
Mistake 1
Incorrect Base Area Calculations
Some students mistakenly calculate the base area incorrectly, especially for complex shapes. Always ensure you use the correct formula for the base shape, whether it's a rectangle, triangle, or other polygon.
Mistake 2
Using Height Instead of Slant Height
In pyramids, students may mistakenly use the vertical height instead of the slant height when calculating lateral surface area. Always use slant height for lateral surface area in pyramids.
Mistake 3
Forgetting to Include All Faces
Students often calculate only some of the faces and forget others in the total surface area. Ensure you account for all faces, including all lateral sides and the base(s).
Mistake 4
Assuming All Prisms and Pyramids are Regular
Some students mistakenly assume all prisms and pyramids are regular (having identical faces or bases), leading to incorrect calculations. Check the shape's properties before applying formulas.
Mistake 5
Solved Examples of Surface Area of Prisms and Pyramids
Find the lateral surface area of a rectangular prism with a base perimeter of 20 cm and a height of 15 cm.
Lateral Surface Area = 300 cm²
Problem 1
Given P = 20 cm, h = 15 cm. Use the formula: Lateral Surface Area = P × h = 20 × 15 = 300 cm²
Find the total surface area of a triangular pyramid with a base perimeter of 18 cm, base area of 24 cm², and slant height of 10 cm.
Explanation
Total Surface Area = 114 cm²
Problem 2
Use the formula: Lateral Surface Area = 1/2 × P × l = 1/2 × 18 × 10 = 90 cm² Total Surface Area = Lateral Surface Area + Base Area = 90 + 24 = 114 cm²
A hexagonal prism has a base area of 30 cm² and height of 12 cm. Find the total surface area, given the perimeter of the base is 24 cm.
Explanation
Total Surface Area = 408 cm²
Problem 3
Find the lateral surface area using: Lateral Surface Area = P × h = 24 × 12 = 288 cm² Use the Total Surface Area formula: Total Surface Area = Lateral Surface Area + 2 × Base Area = 288 + 2 × 30 = 288 + 60 = 348 cm²
Find the lateral surface area of a square pyramid with a base perimeter of 16 cm and slant height of 6 cm.
Explanation
Lateral Surface Area = 48 cm²
Problem 4
Lateral Surface Area = 1/2 × P × l = 1/2 × 16 × 6 = 8 × 6 = 48 cm²
A triangular prism has a height of 10 cm and a base area of 15 cm². Find the total surface area if the base perimeter is 9 cm.
Explanation
Total Surface Area = 210 cm²
It is the total area that covers the outside of the shape, including its lateral sides and the base(s).
1.What are the two types of surface area in prisms and pyramids?
2.What is the difference between slant height and height?
3.How do you calculate the total surface area of a prism?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of Prisms and Pyramids
Students often make mistakes while calculating the surface area of prisms and pyramids, which leads to wrong answers. Below are some common mistakes and the ways to avoid them.
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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.
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