101 LearnersLast updated on August 30, 2025
Surface Area of Pyramid with Triangular Base
A pyramid with a triangular base is a 3-dimensional shape that consists of a triangular base and triangular faces converging to a single vertex. The surface area of such a pyramid is the total area covered by its outer surface. This includes the area of its triangular base and the lateral surface area, which consists of the other triangular faces. In this article, we will learn about the surface area of a pyramid with a triangular base.
What is the Surface Area of a Pyramid with a Triangular Base?
The surface area of a pyramid with a triangular base is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.
This type of pyramid is a 3D shape with a triangular base and triangular faces meeting at a common vertex.
The surface area includes the base area and the lateral surface area, which consists of the triangular faces excluding the base.
Pyramids can have regular or irregular bases, but in this context, we focus on those with a regular triangular base where all edges are equal.
Surface Area of a Pyramid with a Triangular Base Formula
A pyramid with a triangular base has two types of surface areas: the base area and the lateral surface area.
Observe the pyramid below to understand its surface area, height (h), slant height (l), and base side length (a).
A pyramid with a triangular base has two types of surface areas: Base Surface Area Lateral Surface Area
Base Surface Area
The base area of the pyramid is the area of its triangular base.
For a pyramid with a regular triangular base, the formula for the base area (B) is given as:
Base Area = (sqrt(3)/4) × a² square units
Here, a is the length of a side of the triangular base.
Lateral Surface Area
The lateral surface area of the pyramid is the total area of the triangular faces excluding the base.
For a pyramid with a regular triangular base, the formula for the lateral surface area (LSA) is given as:
Lateral Surface Area = (3/2) × a × l square units
Where a is the length of a side of the base, and l is the slant height of the pyramid.
Total Surface Area of a Pyramid with a Triangular Base
The total surface area of the pyramid includes both the base area and the lateral surface area.
It is calculated using the formula:
Total Surface Area = Base Area + Lateral Surface Area
Substituting the formulas: Total Surface Area = (sqrt(3)/4) × a² + (3/2) × a × l
Confusing Base Area with Lateral Surface Area
Students often mix up the base area and lateral surface area calculations. Remember that the base area is for the triangular base only, and the lateral surface area is for the triangular faces excluding the base.
Mistake 1
Using Height Instead of Slant Height
Some students mistakenly use the perpendicular height (h) instead of the slant height (l) when finding the lateral surface area. The slant height is necessary for the lateral surface area formula.
Mistake 2
Miscalculating the Base Area
Mistakes often occur when calculating the base area, especially in applying the formula for a triangular base. Ensure the correct formula is used: (sqrt(3)/4) × a² for a regular triangular base.
Mistake 3
Ignoring the Base Area in Total Surface Area Calculation
Students sometimes forget to include the base area when calculating the total surface area. Always add both the base area and the lateral surface area for the total surface area.
Mistake 4
Assuming All Triangles Are Equilateral
When working with an irregular triangular base, do not assume all sides are equal unless specified. Use the appropriate formula for the given base type.
Mistake 5
Solved Examples of Surface Area of Pyramid with Triangular Base
Find the lateral surface area of a pyramid with a triangular base where each side of the base is 5 cm and the slant height is 8 cm.
LSA = 60 cm²
Problem 1
Given a = 5 cm, l = 8 cm. Use the formula: LSA = (3/2) × a × l = (3/2) × 5 × 8 = 3 × 5 × 4 = 60 cm²
Find the total surface area of a pyramid with a triangular base where each side of the base is 6 cm and the slant height is 10 cm.
Explanation
TSA = 106.18 cm²
Problem 2
Use the formula: Total Surface Area = (sqrt(3)/4) × a² + (3/2) × a × l = (sqrt(3)/4) × 6² + (3/2) × 6 × 10 = (sqrt(3)/4) × 36 + 3 × 6 × 5 = 9sqrt(3) + 90 = 15.588 + 90 = 105.588 cm²
A pyramid has a triangular base with each side measuring 4 cm and a slant height of 7 cm. Find the total surface area.
Explanation
TSA = 49.08 cm²
Problem 3
Find the base area: Base Area = (sqrt(3)/4) × a² = (sqrt(3)/4) × 4² = (sqrt(3)/4) × 16 = 4sqrt(3) Find the lateral surface area: LSA = (3/2) × a × l = (3/2) × 4 × 7 = 3 × 4 × 3.5 = 42 cm² Total Surface Area = Base Area + Lateral Surface Area = 4sqrt(3) + 42 = 6.928 + 42 = 48.928 cm²
Find the lateral surface area of a pyramid with a triangular base where each side of the base is 3 cm and the slant height is 5 cm.
Explanation
LSA = 22.5 cm²
Problem 4
LSA = (3/2) × a × l = (3/2) × 3 × 5 = 3 × 1.5 × 5 = 22.5 cm²
The lateral surface area of a pyramid with a triangular base is 90 cm², and the slant height is 15 cm. Find the side length of the base.
Explanation
Side length = 4 cm
It is the total area that covers the outside of the pyramid, including its triangular base and the lateral surfaces.
1.What are the two types of surface area in a pyramid with a triangular base?
2.What is the difference between slant height and height?
3.What units is surface area measured in?
4.How to calculate the base area of a regular triangular base?
Common Mistakes and How to Avoid Them in the Surface Area of a Pyramid with a Triangular Base
Calculating the surface area of a pyramid with a triangular base can be tricky. Below are some common mistakes and 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.
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