105 LearnersLast updated on August 5th, 2025
Surface Area of Frustum
A frustum is a 3-dimensional shape that results from slicing the top off a cone parallel to its base. The surface area of a frustum includes the areas of its two circular bases and the curved surface that connects them. In this article, we will learn about the surface area of a frustum.
What is the Surface Area of a Frustum?
The surface area of a frustum is the total area occupied by its outer surfaces. It is measured in square units. A frustum is formed by cutting a cone parallel to its base, resulting in two parallel circular bases and a curved surface connecting them. Unlike a complete cone, a frustum does not have a vertex.
Surface Area of a Frustum Formula
A frustum has a curved surface and two circular bases. The surface area of a frustum is the sum of the lateral surface area and the areas of the two bases. Look at the frustum below to see its surface area, height (h), slant height (l), and radii (R and r) of the two bases.
The surface area of a frustum is calculated as: Lateral Surface Area of a Frustum Total Surface Area of a Frustum
Lateral Surface Area of a Frustum
The lateral surface area of a frustum is the area of the curved surface connecting the two circular bases.
It is calculated using the formula: Lateral Surface Area = π(R + r)l square units
Here, R and r are the radii of the larger and smaller bases, respectively, and l is the slant height of the frustum.
Total Surface Area of a Frustum
The total surface area of a frustum includes the lateral surface area and the areas of the two circular bases.
The total surface area is calculated using the formula:
Total Surface Area = Lateral Surface Area + Area of Top Base + Area of Bottom Base = π(R + r)l + πR² + πr²
Where R and r are the radii of the larger and smaller bases, and l is the slant height of the frustum.
Volume of a Frustum
The volume of a frustum is the space enclosed within its surfaces. It can be found using the formula:
Volume = (1/3)πh(R² + r² + Rr) cubic units
Where R and r are the radii of the larger and smaller bases, respectively, and h is the vertical height of the frustum.
Confusion between Lateral Surface Area and Total Surface Area
Students sometimes confuse the lateral surface area with the total surface area of a frustum. Remember, the lateral surface area only includes the curved surface, while the total surface area includes the curved surface and the two bases.
Mistake 1
Using height instead of slant height
Some students mistakenly use the vertical height (h) instead of the slant height (l) for calculating the lateral surface area. Always use the slant height in the formula for lateral surface area: π(R + r)l.
Mistake 2
Using incorrect values for 𝜋
A common mistake is using incorrect values for 𝜋, such as 22. Instead, use accurate values like 22/7 or 3.14.
Mistake 3
Forgetting to include the areas of both bases in the total surface area
Students sometimes calculate only the lateral surface area and forget to add the areas of the two bases. Always include all parts when calculating the total surface area.
Mistake 4
Assuming the lateral surface area and curved surface area are different
Some students mistakenly believe that lateral surface area and curved surface area are different, but in a frustum, they refer to the same part. Use the same formula for both.
Mistake 5
Solved Examples of Surface Area of Frustum
Find the lateral surface area of a frustum with larger base radius 8 cm, smaller base radius 5 cm, and slant height 12 cm.
Lateral Surface Area = 490.08 cm²
Problem 1
Given R = 8 cm, r = 5 cm, l = 12 cm. Use the formula: Lateral Surface Area = π(R + r)l = 3.14 × (8 + 5) × 12 = 3.14 × 13 × 12 = 490.08 cm²
Find the total surface area of a frustum with larger base radius 6 cm, smaller base radius 4 cm, and slant height 10 cm.
Explanation
Total Surface Area = 628.32 cm²
Problem 2
Use the formula: Total Surface Area = π(R + r)l + πR² + πr² = 3.14 × (6 + 4) × 10 + 3.14 × 6² + 3.14 × 4² = 3.14 × 10 × 10 + 3.14 × 36 + 3.14 × 16 = 314 + 113.04 + 50.24 = 628.32 cm²
A frustum has a larger base radius of 7 cm, a smaller base radius of 3 cm, and a height of 9 cm. Find the total surface area.
Explanation
Total Surface Area = 579.58 cm²
Problem 3
First, find the slant height using the Pythagorean theorem: l = √((R - r)² + h²) = √((7 - 3)² + 9²) = √(16 + 81) = √97 = 9.85 cm Use the formula: Total Surface Area = π(R + r)l + πR² + πr² = 3.14 × (7 + 3) × 9.85 + 3.14 × 7² + 3.14 × 3² = 3.14 × 10 × 9.85 + 3.14 × 49 + 3.14 × 9 = 309.41 + 153.86 + 28.26 = 579.58 cm²
Find the lateral surface area of a frustum with larger base radius 10 cm, smaller base radius 6 cm, and slant height 8 cm.
Explanation
Lateral Surface Area = 402.12 cm²
Problem 4
Lateral Surface Area = π(R + r)l = 3.14 × (10 + 6) × 8 = 3.14 × 16 × 8 = 402.12 cm²
The slant height of a frustum is 14 cm, its larger base radius is 9 cm, and its lateral surface area is 792 cm². Find the smaller base radius.
Explanation
Smaller Base Radius = 7 cm
It is the total area that covers the outside of the frustum, including its curved side and the two bases.
1.What are the components of the surface area of a frustum?
2.What is the difference between slant height and height in a frustum?
3.Are lateral surface area and curved surface area the same in a frustum?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Frustum
Students often make mistakes while calculating the surface area of a frustum, 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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