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105 LearnersLast updated on September 4, 2025
Surface Area of Triangular Pyramid
A triangular pyramid, also known as a tetrahedron, is a 3-dimensional shape with a triangular base and three triangular faces that meet at a common point called the apex. The surface area of a triangular pyramid is the total area covered by its outer surface, including all its faces. In this article, we will learn about the surface area of a triangular pyramid.
What is the Surface Area of a Triangular Pyramid?
The surface area of a triangular pyramid is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.
A triangular pyramid has four faces: one triangular base and three triangular lateral faces. The surface area is calculated by adding the area of the base and the areas of the lateral faces.
Triangular pyramids can be regular, where all faces are equilateral triangles, or irregular, where the faces can be any type of triangle.
Surface Area of a Triangular Pyramid Formula
A triangular pyramid consists of a base and lateral faces, and its surface area includes both. Look at the triangular pyramid below to see its surface area, base area, and lateral surface areas.
A triangular pyramid has two types of surface areas:
- Base Area of a Triangular Pyramid
- Total Surface Area of a Triangular Pyramid
Base Area of a Triangular Pyramid
The base area of a triangular pyramid is simply the area of its triangular base. The formula to find the area of a triangle depends on the type of triangle.
For an equilateral triangle, the area is (sqrt(3)/4) * side2.
For other triangles, Heron's formula or the traditional base-height formula (1/2 * base * height) can be used.
Total Surface Area of a Triangular Pyramid
The total surface area of a triangular pyramid is the sum of the base area and the areas of the three lateral triangular faces.
If the side lengths of the base and the slant heights of the faces are known, you can calculate the total surface area by finding the area of each face and adding them together.
Volume of a Triangular Pyramid
The volume of a triangular pyramid indicates how much space is inside it.
The volume can be calculated using the formula: Volume = (1/3) * Base Area * Height
Where the height is the perpendicular distance from the base to the apex.
Confusion between Base Area and Total Surface Area
Students assume that the base area and the total surface area of a triangular pyramid are the same. This confusion arises because both involve the dimensions of the base. Always remember that the base area is just one part of the surface area, and the total surface area includes all faces.
Mistake 1
Using Incorrect Formulas for Triangle Area
Some students mistakenly use incorrect formulas for finding the area of the triangular base or faces. Ensure the correct formula is used for the type of triangle you are dealing with.
Mistake 2
Neglecting Lateral Faces
A common mistake is neglecting to include the lateral faces in the total surface area calculation. Always consider the areas of all triangular faces when calculating the total surface area.
Mistake 3
Assuming All Faces are Equilateral
Some students mistakenly believe that all triangular faces are equilateral. If the dimensions vary, ensure you calculate the area for each face individually.
Mistake 4
Incorrectly Calculating Slant Heights
Another mistake is using incorrect slant heights for the lateral faces. Make sure the slant height is measured along the face of the pyramid and not along the vertical height.
Mistake 5
Solved Examples of Surface Area of Triangular Pyramid
Find the total surface area of a triangular pyramid with a base area of 9 cm² and lateral face areas of 12 cm², 15 cm², and 18 cm².
Total Surface Area = 54 cm²
Problem 1
Add the base area and the lateral face areas: Total Surface Area = Base Area + Sum of Lateral Face Areas = 9 cm² + 12 cm² + 15 cm² + 18 cm² = 54 cm²
A triangular pyramid has an equilateral base with a side length of 6 cm. The slant height of each lateral face is 5 cm. Find the total surface area.
Explanation
Total Surface Area = 78 cm²
Problem 2
Calculate the base area using the formula for an equilateral triangle: Base Area = (sqrt(3)/4) * side² = (sqrt(3)/4) * 6² = 9sqrt(3) cm² Calculate the area of one lateral face: Lateral Face Area = (1/2) * base * slant height = (1/2) * 6 * 5 = 15 cm² Total Surface Area = Base Area + 3 * Lateral Face Area = 9sqrt(3) cm² + 3 * 15 cm² = 9sqrt(3) cm² + 45 cm² = approximately 78 cm²
A triangular pyramid has a right triangular base with legs of 3 cm and 4 cm. The slant heights of the lateral faces are 5 cm, 6 cm, and 7 cm. Find the total surface area.
Explanation
Total Surface Area = 69 cm²
Problem 3
Calculate the base area using the formula for a right triangle: Base Area = (1/2) * base * height = (1/2) * 3 * 4 = 6 cm² Calculate the area of each lateral face: Lateral Face 1 Area = (1/2) * 3 * 5 = 7.5 cm² Lateral Face 2 Area = (1/2) * 4 * 6 = 12 cm² Lateral Face 3 Area = (1/2) * 5 * 7 = 17.5 cm² Total Surface Area = Base Area + Sum of Lateral Face Areas = 6 cm² + 7.5 cm² + 12 cm² + 17.5 cm² = 69 cm²
Find the surface area of a triangular pyramid with a base area of 10 cm² and three equal lateral faces each with an area of 8 cm².
Explanation
Total Surface Area = 34 cm²
Problem 4
Add the base area and the lateral face areas: Total Surface Area = Base Area + 3 * Lateral Face Area = 10 cm² + 3 * 8 cm² = 10 cm² + 24 cm² = 34 cm²
The slant heights of a triangular pyramid's lateral faces are 9 cm, 10 cm, and 11 cm, and their corresponding base sides are 4 cm, 5 cm, and 6 cm. Find the total lateral surface area.
Explanation
Total Lateral Surface Area = 136 cm²
It is the total area that covers the outside of the triangular pyramid, including its base and all lateral triangular faces.
1.What are the two types of surface area in a triangular pyramid?
2.How do you calculate the base area of an equilateral triangle?
3.What is the difference between slant height and height in a triangular pyramid?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Triangular Pyramid
Students often make mistakes while calculating the surface area of a triangular pyramid, which leads to incorrect 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
