106 LearnersLast updated on August 5th, 2025
Surface Area of Right Pyramid
A right pyramid is a 3-dimensional shape with a polygonal base and triangular faces that meet at a common point, called the apex. The surface area of a right pyramid is the total area covered by its outer surface. This includes both the lateral surface area (the sum of the areas of the triangular faces) and the base area. In this article, we will learn about the surface area of a right pyramid.
What is the Surface Area of a Right Pyramid?
The surface area of a right pyramid is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.
A right pyramid is a 3D shape with a polygonal base and triangular faces that meet at a single point called the apex. The apex is directly above the center of the base, making the pyramid symmetrical.
The surface area of a right pyramid includes both the lateral surface area (the triangular faces) and the base area.
Surface Area of a Right Pyramid Formula
A right pyramid has a lateral surface, and it has two types of surface areas: the lateral surface area and the total surface area.
The lateral surface area is the sum of the areas of the triangular faces, while the total surface area includes both the lateral surface area and the base area.
A right pyramid has two types of surface areas: Lateral Surface Area of a Right Pyramid Total Surface Area of a Right Pyramid
Lateral Surface Area of a Right Pyramid
The lateral surface area of a right pyramid is the sum of the areas of the triangular faces that connect the base to the apex. The formula for the lateral surface area depends on the shape of the base.
For a pyramid with a regular polygon base, the formula is: Lateral Surface Area = 1/2 × Perimeter × Slant Height
Here, the perimeter is the perimeter of the base of the pyramid. The slant height is the height of each triangular face from the base to the apex.
Total Surface Area of a Right Pyramid
The total surface area of a right pyramid is the sum of the lateral surface area and the base area.
The formula is: Total Surface Area = Lateral Surface Area + Base Area
Derivation of the Total Surface Area of a Right Pyramid
To find the total surface area of a right pyramid, calculate the lateral surface area using the perimeter and slant height, then add the area of the base.
Let us consider a right pyramid with a regular polygon base.
Total surface area of a right pyramid = base area + lateral surface area
Lateral surface area = 1/2 × perimeter × slant height
Substituting the formulas into the total surface area, Total surface area = base area + 1/2 × perimeter × slant height
Volume of a Right Pyramid
The volume of a right pyramid shows how much space is inside it. It tells us how much space is inside the pyramid or how much it can hold.
The volume of a right pyramid can be found by using the formula: Volume = 1/3 × Base Area × Height (cubic unit)
Confusion between Lateral Surface Area and Total Surface Area
Students assume that the lateral surface area and the total surface area of a right pyramid are the same. This confusion arises because both involve the slant height. Always remember that the lateral surface area is used only for the triangular faces, while the total surface area includes both the triangular faces and the base.
Mistake 1
Using height instead of slant height
Some students mistakenly use the vertical height instead of the slant height when finding the lateral surface area. Remember the formula for lateral surface area is 1/2 × perimeter × slant height, so always use the slant height.
Mistake 2
Incorrect calculation of the base perimeter
Students may miscalculate the perimeter of the base, especially if the base is not a simple shape like a square. Ensure you correctly add the lengths of all sides of the base to find the perimeter.
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 base area. Always include both parts when calculating the total surface area.
Total Surface Area = Lateral Surface Area + Base Area.
Mistake 4
Assuming all triangular faces are the same
Some students mistakenly believe that all triangular faces in a right pyramid are the same, which may not be true if the base is not regular. Ensure you calculate each triangle's area correctly.
Mistake 5
Solved Examples of Surface Area of Right Pyramid
Find the lateral surface area of a right pyramid with a square base where each side is 6 cm and the slant height is 10 cm.
Lateral Surface Area = 120 cm²
Problem 1
Given the side of the square base = 6 cm, slant height = 10 cm. Perimeter = 4 × 6 = 24 cm. Use the formula: Lateral Surface Area = 1/2 × Perimeter × Slant Height = 1/2 × 24 × 10 = 12 × 10 = 120 cm²
Find the total surface area of a right pyramid with a triangular base with sides 5 cm each and a slant height of 8 cm.
Explanation
Total Surface Area = 65.48 cm²
Problem 2
Perimeter of the base = 3 × 5 = 15 cm. Lateral Surface Area = 1/2 × 15 × 8 = 60 cm². Base Area (for an equilateral triangle) = (√3/4) × 5² = 10.83 cm². Total Surface Area = Lateral Surface Area + Base Area = 60 + 10.83 = 70.83 cm²
A right pyramid has a rectangular base with dimensions 4 cm by 6 cm and a slant height of 9 cm. Find the total surface area.
Explanation
Total Surface Area = 130 cm²
Problem 3
Perimeter of the base = 2(4 + 6) = 20 cm. Lateral Surface Area = 1/2 × 20 × 9 = 90 cm². Base Area = 4 × 6 = 24 cm². Total Surface Area = Lateral Surface Area + Base Area = 90 + 24 = 114 cm²
Find the lateral surface area of a right pyramid with a hexagonal base where each side is 3 cm and the slant height is 7 cm.
Explanation
Lateral Surface Area = 63 cm²
Problem 4
Perimeter of the hexagonal base = 6 × 3 = 18 cm. Use the formula: Lateral Surface Area = 1/2 × Perimeter × Slant Height = 1/2 × 18 × 7 = 9 × 7 = 63 cm²
The slant height of a right pyramid is 12 cm, and its lateral surface area is 180 cm². Find the perimeter of the base.
Explanation
Perimeter = 30 cm
It is the total area that covers the outside of a right pyramid, including its triangular faces and the base.
1.What are the two types of surface area in a right pyramid?
2.What is the difference between slant height and height?
3.How do you find the lateral surface area of a right pyramid?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Right Pyramid
Students often make mistakes while calculating the surface area of a right 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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: She has songs for each table which helps her to remember the tables
