102 LearnersLast updated on August 30, 2025
Surface Area of Right Square Pyramid
A right square pyramid is a 3-dimensional shape that has a square base. The surface area of a right square pyramid is the total area covered by its outer surface. The surface area of the right square pyramid includes both its lateral surface and its base. In this article, we will learn about the surface area of a right square pyramid.
What is the Surface Area of a Right Square Pyramid?
The surface area of a right square pyramid is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.
A right square pyramid is a 3D shape with a square base and triangular faces that meet at a common point called the apex.
It has a flat square base and slanted triangular faces, so it has two surface areas: the lateral surface area and the total surface area.
Right square pyramids are named for having an apex directly above the center of the square base.
Surface Area of a Right Square Pyramid Formula
A right square pyramid has a lateral surface area and a total surface area.
Look at the pyramid below to see its surface area, height (h), slant height (l), and side length of the base (a).
A right square pyramid has two types of surface areas:
Lateral Surface Area of a Right Square Pyramid
Total Surface Area of a Right Square Pyramid
Lateral Surface Area of a Right Square Pyramid
The area of the triangular faces of the pyramid, excluding its base, is known as the lateral surface area of a right square pyramid.
The formula for the lateral surface area (LSA) of the right square pyramid is given as:
Lateral Surface Area = 2al square units
Here, a is the side length of the base. l is the slant height of the pyramid.
Total Surface Area of a Right Square Pyramid
The total area occupied by the pyramid, including the area of the lateral surface and the area of the square base, is known as the total surface area of the right square pyramid.
The total surface area is calculated by using the formula:
Total surface area = a² + 2al square units Where a is the side length of the base. l is the slant height of the pyramid.
Derivation of the Total Surface Area of a Right Square Pyramid To find the total surface area of a right square pyramid, consider the base and the triangular faces.
The base area is simply the area of the square base, a².
The lateral surface area is the sum of the areas of the four triangular faces.
Each triangular face has an area of (1/2)al.
Therefore, the lateral surface area is 4 × (1/2)al = 2al.
Total surface area of a right square pyramid = base area + lateral surface area
Here, base area = a² Lateral surface area = 2al
Substituting the formulas into the total surface area, Total surface area, T = a² + 2al
Volume of a Right Square Pyramid
The volume of a right square pyramid shows how much space is inside it. It tells us how much space the pyramid can hold.
The volume of a right square pyramid can be found by using the formula: Volume = (1/3)a²h cubic units
Confusion between LSA and TSA
Students assume that the lateral surface area (LSA) and the total surface area (TSA) of a right square pyramid are the same.
This confusion arises because both involve the slant height and the base.
Always remember that LSA includes only the triangular faces, while TSA includes the base as well.
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 2al, so always use the slant height.
Mistake 2
Misidentifying the base as a different shape
A common mistake is assuming the base of the pyramid is not a square. Always verify that the base is indeed a square with equal side lengths for accurate calculations.
Mistake 3
Forgetting to include the base area in the total surface area
Students often calculate only the lateral surface and forget to add the square base. Always include both parts when calculating the total surface area. Total surface area = lateral surface area + base area.
Mistake 4
Assuming lateral surface area and triangular face area are different
Some students mistakenly believe that the lateral surface area and the area of the triangular faces are different, but in a right square pyramid, they are the same. So, use the same formula for both.
Mistake 5
Solved Examples of Surface Area of Right Square Pyramid
Find the lateral surface area of a right square pyramid with a base side length of 6 cm and a slant height of 10 cm.
LSA = 120 cm²
Problem 1
Given a = 6 cm, l = 10 cm. Use the formula: LSA = 2al = 2 × 6 × 10 = 12 × 10 = 120 cm²
Find the total surface area of a right square pyramid with a base side length of 5 cm and a slant height of 13 cm.
Explanation
TSA = 155 cm²
Problem 2
Use the formula: TSA = a² + 2al = 5² + 2 × 5 × 13 = 25 + 130 = 155 cm²
A right square pyramid has a base side length of 4 cm and a height of 9 cm.
Find the total surface area.
Explanation
TSA = 88.94 cm²
Problem 3
Find the slant height using: l = √(h² + (a/2)²) = √(9² + 2²) = √(81 + 4) = √85 ≈ 9.22 cm Use the TSA formula: TSA = a² + 2al = 4² + 2 × 4 × 9.22 = 16 + 73.76 ≈ 88.94 cm²
Find the lateral surface area of a right square pyramid with a base side length of 3.5 cm and a slant height of 8 cm.
Explanation
LSA = 56 cm²
Problem 4
LSA = 2al = 2 × 3.5 × 8 = 7 × 8 = 56 cm²
The slant height of a right square pyramid is 12 cm, and its lateral surface area is 96 cm². Find the base side length.
Explanation
Base side length = 4 cm
It is the total area that covers the outside of the pyramid, including its triangular faces and the square base.
1.What are the two types of surface area in a right square pyramid?
2.What is the difference between slant height and height?
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 Right Square Pyramid
Students often make mistakes while calculating the surface area of a right square 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
