114 LearnersLast updated on August 5th, 2025
Surface Area of Regular Pyramid
A regular pyramid is a three-dimensional shape with a polygonal base and triangular faces that meet at a common vertex. The surface area of a regular pyramid is the total area covered by its outer surface, which includes both the base area and the lateral (triangular) faces. In this article, we will explore how to calculate the surface area of a regular pyramid.
What is the Surface Area of a Regular Pyramid?
The surface area of a regular pyramid is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.
A regular pyramid has a base that is a regular polygon (all sides and angles are equal) and triangular faces that are congruent.
The surface area consists of the base area and the lateral surface area, which is the sum of the areas of the triangular faces.
Surface Area of a Regular Pyramid Formula
A regular pyramid has a base and lateral triangular faces. The surface area is the sum of the area of the base and the lateral surface area.
The lateral surface area can be calculated by multiplying the perimeter of the base by the slant height and then dividing by 2.
A regular pyramid has two parts of surface areas: Base Area of a Regular Pyramid Lateral Surface Area of a Regular Pyramid
Base Area of a Regular Pyramid
The base area of a regular pyramid depends on the shape of the base.
For example, if the base is a square with side length "a," the base area is a². For other regular polygons with "n" sides and side length "s," the base area can be calculated using specific formulas for each polygon.
Lateral Surface Area of a Regular Pyramid
The lateral surface area of a regular pyramid is the total area of its triangular faces.
It can be calculated using the formula: Lateral Surface Area = (Perimeter of the base × Slant height) / 2
The slant height is the distance from the vertex to the midpoint of a side of the base.
Volume of a Regular Pyramid
The volume of a regular pyramid is the amount of space enclosed within it.
It is given by the formula: Volume = (1/3) × Base Area × Height
where the height is the perpendicular distance from the base to the vertex.
Confusion between Base Area and Lateral Surface Area
Students sometimes confuse the base area and the lateral surface area. Always remember that the base area is the area of the polygonal base, while the lateral surface area includes the triangular faces.
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. The formula requires the slant height, so it is crucial to distinguish between the two.
Mistake 2
Incorrect Calculation of the Perimeter
A common mistake is miscalculating the perimeter of the base, especially for polygons with many sides. Ensure every side is accounted for and measured correctly.
Mistake 3
Forgetting to Include the Base Area in Total Surface Area
Students often calculate only the lateral surface area and forget to add the base area. Always include both parts when calculating the total surface area.
Mistake 4
Assuming All Triangles Are the Same in Irregular Pyramids
In irregular pyramids, the triangular faces may not be congruent. Always check the shape of the base and the congruency of the triangles before assuming all are the same.
Mistake 5
Solved Examples of Surface Area of Regular Pyramid
Find the lateral surface area of a regular pyramid with a square base of side 6 cm and a slant height of 10 cm.
Lateral Surface Area = 120 cm²
Problem 1
Given a square base with side 6 cm and slant height 10 cm. Perimeter = 4 × 6 = 24 cm Use the formula: Lateral Surface Area = (Perimeter × Slant height) / 2 = (24 × 10) / 2 = 240 / 2 = 120 cm²
Find the total surface area of a regular pyramid with a hexagonal base with a side length of 4 cm and a slant height of 9 cm.
Explanation
Total Surface Area = 156.56 cm²
Problem 2
Hexagonal base: Perimeter = 6 × 4 = 24 cm Base Area (hexagon) = (3√3/2) × side² = (3√3/2) × 16 ≈ 41.57 cm² Lateral Surface Area = (24 × 9) / 2 = 108 cm² Total Surface Area = Base Area + Lateral Surface Area ≈ 41.57 + 108 = 149.57 cm²
A regular pyramid has a triangular base with sides 5 cm each and a slant height of 6 cm. Find the total surface area.
Explanation
Total Surface Area ≈ 81.65 cm²
Problem 3
Triangular base: Perimeter = 3 × 5 = 15 cm Base Area = (√3/4) × side² = (√3/4) × 25 ≈ 10.83 cm² Lateral Surface Area = (15 × 6) / 2 = 45 cm² Total Surface Area = Base Area + Lateral Surface Area ≈ 10.83 + 45 = 55.83 cm²
Find the lateral surface area of a regular pyramid with a pentagonal base, each side measuring 4 cm, and a slant height of 7 cm.
Explanation
Lateral Surface Area = 70 cm²
Problem 4
Pentagonal base: Perimeter = 5 × 4 = 20 cm Use the formula: Lateral Surface Area = (Perimeter × Slant height) / 2 = (20 × 7) / 2 = 140 / 2 = 70 cm²
The total surface area of a regular pyramid is 200 cm², and its lateral surface area is 150 cm². Find the base area.
Explanation
Base Area = 50 cm²
It is the total area that covers the outside of the pyramid, including its base and the lateral triangular faces.
1.What are the two components of surface area in a regular pyramid?
2.What is the difference between slant height and height in a pyramid?
3.Is the lateral surface area the same for all regular pyramids?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Regular Pyramid
Students often make mistakes while calculating the surface area of a regular pyramid, leading to incorrect results. Below are some common mistakes and 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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