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102 LearnersLast updated on September 4, 2025
Surface Area of Pyramid with Square Base
A pyramid with a square base is a 3-dimensional shape consisting of a square base and four triangular faces. The surface area of such a pyramid is the total area covered by its outer surface. This includes the area of the square base and the areas of the triangular faces. In this article, we will learn about the surface area of a pyramid with a square base.
What is the Surface Area of a Pyramid with a Square Base?
The surface area of a pyramid with a square base is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.
A pyramid with a square base has a flat base that is a square and triangular sides that meet at a point called the apex or vertex.
The surface area includes both the base area and the lateral surface area. The lateral surface area consists of the four triangles.
Surface Area of a Pyramid with Square Base Formula
A pyramid with a square base has two main components contributing to its surface area: the base area and the lateral surface area. Base Area of the Pyramid:
The base is a square, and its area is given by the formula: Base Area = side2
Lateral Surface Area of the Pyramid: The lateral surface area is the sum of the areas of the four triangular faces. If the slant height (l) and the base side length (s) are known, then each triangular face has an area of:
Area of one triangle = 1/2 x s x l
Thus, the total lateral surface area is:
Lateral Surface Area = 4 \times \left(\frac{1}{2} \times s \times l\right) = 2sl \]
Total Surface Area of the Pyramid: Total Surface Area = Base Area + Lateral Surface Area = s2 + 2sl
Lateral Surface Area of a Pyramid with Square Base
The lateral surface area of a pyramid with a square base consists of the four triangular faces excluding the base. Each triangle's area is calculated using the base of the triangle (side of the square base) and the slant height of the pyramid.
The formula for the lateral surface area is:
Lateral Surface Area = 2sl
Here, s is the side length of the square base, and l is the slant height of the pyramid.
Total Surface Area of a Pyramid with Square Base
The total surface area of a pyramid with a square base includes the base area and the lateral surface area. Using the formulas for the base area and the lateral surface area, the total surface area can be found as follows:
Total Surface Area = s2 + 2sl Where s is the side length of the base, and l is the slant height of the pyramid.
Volume of a Pyramid with Square Base
The volume of a pyramid with a square base represents the amount of space it occupies. It is one-third of the volume of a prism with the same base area and height.
The formula for the volume of a pyramid with a square base is given by: Volume = 1/3 x s2 x h where s is the side length of the base, and h is the vertical height from the base to the apex.
Confusion between Lateral Surface Area and Total Surface Area
Students assume that the lateral surface area and the total surface area of a pyramid are the same. Always remember that the lateral surface area includes only the triangular faces, while the total surface area includes both the triangular faces and the square base.
Mistake 1
Using Height Instead of Slant Height
Some students mistakenly use the vertical height instead of the slant height when calculating the lateral surface area. Remember, the formula for lateral surface area is 2sl, so always use the slant height.
Mistake 2
Incorrect Calculation of Base Area
A common mistake is using incorrect values or formulas for calculating the base area. Ensure you use the correct formula: Base Area = s^2, where s is the side length of the square base.
Mistake 3
Forgetting to Include All Triangular Faces in the Lateral Surface Area
Students sometimes miscalculate the lateral surface area by not accounting for all four triangular faces. Always remember to compute the area for all four triangles: Lateral Surface Area = 4 × (1/2 × s × l) = 2sl.
Mistake 4
Assuming Different Formulas for Lateral and Total Surface Area
Some students mistakenly believe that lateral surface area and total surface area require different base area calculations. They don't; the distinction is only in whether the base area is included.
Mistake 5
Solved Examples of Surface Area of Pyramid with Square Base
Find the lateral surface area of a pyramid with a square base of side 8 cm and a slant height of 10 cm.
Lateral Surface Area = 160 cm²
Problem 1
Given s = 8 cm, l = 10 cm. Use the formula: Lateral Surface Area = 2sl = 2 × 8 × 10 = 160 cm²
Find the total surface area of a pyramid with a square base with side 5 cm and slant height 13 cm.
Explanation
Total Surface Area = 155 cm²
Problem 2
Use the formula: Total Surface Area = s² + 2sl = 5² + 2 × 5 × 13 = 25 + 130 = 155 cm²
A pyramid has a square base of side 6 cm and a height of 8 cm. Find the total surface area.
Explanation
Total Surface Area = 180 cm²
Problem 3
Find the slant height using: l = √(h² + (s/2)²) = √(8² + (6/2)²) = √(64 + 9) = √73 ≈ 8.54 cm Use the formula: Total Surface Area = s² + 2sl = 6² + 2 × 6 × 8.54 = 36 + 102.48 ≈ 138.48 cm²
Find the lateral surface area of a pyramid with a square base of side 3.5 cm and slant height 5 cm.
Explanation
Lateral Surface Area = 35 cm²
Problem 4
Lateral Surface Area = 2sl = 2 × 3.5 × 5 = 35 cm²
The slant height of a pyramid is 15 cm, and its lateral surface area is 660 cm². Find the side length of the base.
Explanation
Side Length = 22 cm
It is the total area that covers the outside of the pyramid, including its four triangular faces and the square base.
1.What are the two types of surface area in a pyramid with a square base?
2.What is the difference between slant height and height?
3.Is lateral surface area the same as total surface area?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Pyramid with Square Base
Students often make mistakes while calculating the surface area of a pyramid with a square base, which leads to incorrect 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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: She has songs for each table which helps her to remember the tables
