Last updated on July 31st, 2025
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.
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.
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: -
Pyramids have: -
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
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
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
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).
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.
Total Surface Area = 114 cm²
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.
Total Surface Area = 408 cm²
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.
Lateral Surface Area = 48 cm²
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.
Total Surface Area = 210 cm²
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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