Surface Area of Right Square Pyramid
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Last updated on August 30, 2025

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.

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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.

Problem 1

Given a = 6 cm, l = 10 cm. Use the formula: LSA = 2al = 2 × 6 × 10 = 12 × 10 = 120 cm²

Okay, lets begin

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

Well explained 👍

Problem 2

Use the formula: TSA = a² + 2al = 5² + 2 × 5 × 13 = 25 + 130 = 155 cm²

Okay, lets begin

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

Well explained 👍

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

Okay, lets begin

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.

Well explained 👍

Problem 4

LSA = 2al = 2 × 3.5 × 8 = 7 × 8 = 56 cm²

Okay, lets begin

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

Well explained 👍

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?

Lateral surface area and total surface area are the two types of surface area in a right square pyramid.

2.What is the difference between slant height and height?

Slant height is the length from the apex to the midpoint of a base edge. Height is the perpendicular distance from the apex to the center of the base.

3.Is lateral surface area the same as the area of the triangular faces?

Yes, in right square pyramids, both lateral surface area and the area of the triangular faces mean the same.

4.What unit is surface area measured in?

Surface area is always measured in square units like cm², m², or in².

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.

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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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