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103 LearnersLast updated on September 4, 2025
Surface Area of Square Pyramid
A square pyramid is a 3-dimensional shape with a square base and triangular sides that meet at a point. The surface area of a square pyramid is the total area covered by its outer surface. The surface area includes both the base and the lateral (triangular) surfaces. In this article, we will learn about the surface area of a square pyramid.
What is the Surface Area of a Square Pyramid?
The surface area of a square pyramid is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.
A square pyramid has a square base and four triangular faces that meet at a common point known as the apex.
The surface area consists of the base area and the lateral surface area, which is the sum of the areas of all the triangular faces.
Surface Area of a Square Pyramid Formula
A square pyramid has a base and four triangular surfaces, contributing to two types of surface areas: the base area and the total surface area.
Look at the square pyramid below to see its surface area, base length (b), slant height (l), and height (h).
A square pyramid has two key components in its surface area calculation:
- Base Area of a Square Pyramid
- Total Surface Area of a Square Pyramid
Base Area of a Square Pyramid
The area of the base of the square pyramid is simply the area of the square.
The formula for the base area is: Base Area = b² square units
Here, b is the length of the side of the square base.
Total Surface Area of a Square Pyramid
The total area occupied by the square pyramid, including the base area and the lateral surface area, is known as the total surface area.
The total surface area is calculated by using the formula: Total Surface Area = Base Area + Lateral Surface Area
The lateral surface area of a square pyramid is the sum of the areas of its four triangular faces. The formula for the lateral surface area is:
Lateral Surface Area = 2bl square units Where b is the base length, and l is the slant height of the pyramid.
Thus, the total surface area is given by: Total Surface Area = b² + 2bl
Volume of a Square Pyramid
The volume of a square pyramid represents the amount of space enclosed within it. It tells us how much space is inside the pyramid or how much it can hold.
The volume formula for a square pyramid is: Volume = (1/3) × b² × h (cubic units) Where b is the base length, and h is the height of the pyramid.
Confusion between Base Area and Total Surface Area
Students assume that the base area and the total surface area of a square pyramid are the same. This confusion arises because both involve the side length of the base. Always remember that the total surface area includes both the base area and the lateral surface area.
Mistake 1
Using Height instead of Slant Height
Some students mistakenly use the vertical height (h) instead of the slant height (l) when finding the lateral surface area. Remember the formula for lateral surface area is 2bl, so always use the slant height.
Mistake 2
Using the wrong value for calculations
A common mistake of students is miscalculating multiplication or using incorrect units. Always pay attention to units and calculations to ensure accuracy.
Mistake 3
Forgetting to include both the base and lateral areas in the total surface area
Students often calculate only the lateral surface and forget to add the base area. Always include both parts when calculating the total surface area. Total surface area = base area + lateral surface area.
Mistake 4
Assuming all faces are equal without checking dimensions
Some students mistakenly believe that all triangular faces are equal, leading to errors. Always verify the dimensions of each face when calculating surface areas.
Mistake 5
Solved Examples of Surface Area of Square Pyramid
Find the total surface area of a square pyramid with a base length of 5 cm and a slant height of 10 cm.
TSA = 125 cm²
Problem 1
Given b = 5 cm, l = 10 cm. Use the formula: Total Surface Area = b² + 2bl = 5² + 2 × 5 × 10 = 25 + 100 = 125 cm²
Calculate the total surface area of a square pyramid with a base length of 8 cm and slant height of 15 cm.
Explanation
TSA = 304 cm²
Problem 2
Use the formula: Total Surface Area = b² + 2bl = 8² + 2 × 8 × 15 = 64 + 240 = 304 cm²
A square pyramid has a base length of 6 cm and a height of 8 cm. Find the total surface area.
Explanation
TSA = 180 cm²
Problem 3
First, find the slant height using the Pythagorean theorem: l = √(h² + (b/2)²) = √(8² + (6/2)²) = √(64 + 9) = √73 ≈ 8.54 cm Use the TSA formula: Total Surface Area = b² + 2bl = 6² + 2 × 6 × 8.54 = 36 + 102.48 = 138.48 cm²
Find the lateral surface area of a square pyramid with a base length of 4 cm and slant height of 7 cm.
Explanation
LSA = 56 cm²
Problem 4
LSA = 2bl = 2 × 4 × 7 = 56 cm²
The slant height of a square pyramid is 12 cm, and its lateral surface area is 96 cm². Find the base length.
Explanation
Base Length = 4 cm
It is the total area that covers the outside of the square pyramid, including its base and the lateral (triangular) surfaces.
1.What are the two types of surface area in a square pyramid?
2.What is the difference between slant height and height?
3.Do all faces of a square pyramid have the same area?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Square Pyramid
Students often make mistakes while calculating the surface area of a square pyramid, which leads to incorrect answers. Below are some common mistakes and ways to avoid them.
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Seyed Ali Fathima S
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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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