102 LearnersLast updated on July 31st, 2025
Surface Area of Frustum of Cone
A frustum of a cone is a 3-dimensional shape that results from slicing a cone parallel to its base. The surface area of the frustum of a cone is the total area covered by its outer surface. The surface area includes both its curved surface and its two circular bases. In this article, we will learn about the surface area of a frustum of a cone.
What is the Surface Area of the Frustum of a Cone?
The surface area of a frustum of a cone is the total area occupied by the boundary or surface of the frustum. It is measured in square units.
A frustum is formed when a cone is sliced parallel to its base, resulting in a shape with two circular bases of different radii and a curved lateral surface.
The frustum has a top and bottom base, unlike the complete cone, and its surface area includes the curved surface area and the areas of both circular bases.
Surface Area of a Frustum of a Cone Formula
A frustum of a cone has a curved surface and two circular bases, and its surface area is classified into the curved surface area and the total surface area.
Observe the frustum below to see its surface area, height (h), slant height (l), and the radii of the top (R) and bottom (r) bases.
A frustum has the following surface areas: Curved Surface Area of a Frustum Total Surface Area of a Frustum
Curved Surface Area of a Frustum
The area of the curved part of the frustum, excluding its bases, is known as the curved surface area of a frustum.
The formula for the CSA (Curved Surface Area) of the frustum is given as: Curved Surface Area = π(R + r)l square units
Here, R is the radius of the top base of the frustum, r is the radius of the bottom base of the frustum, and l is the slant height of the frustum.
Total Surface Area of a Frustum
The total area occupied by the frustum, including the area of the curved surface and the areas of the two circular bases, is known as the total surface area of the frustum.
The total surface area is calculated using the formula: Total Surface Area = π(R + r)l + πR² + πr² square units
Where R is the radius of the top base, r is the radius of the bottom base, and l is the slant height of the frustum.
Derivation of the Total Surface Area of a Frustum
To find the total surface area of a frustum, consider its geometry, which includes the two circular bases and the curved surface.
Total surface area of a frustum = base area of the top circle + base area of the bottom circle + curved surface area of the frustum
Here, base area of top circle = πR²
base area of bottom circle = πr²
Curved surface area of a frustum = π(R + r)l
Substituting the formulas into the total surface area,
Total surface area of a frustum, T = πR² + πr² + π(R + r)l
Therefore, the total surface area of the frustum T = π(R² + r² + (R + r)l)
Volume of a Frustum
The volume of a frustum shows how much space is inside it. It tells us how much space the frustum can hold. The volume of a frustum can be found using the formula: Volume = (1/3)πh(R² + r² + Rr) (cubic units) Where h is the height of the frustum, R is the radius of the top base, and r is the radius of the bottom base.
Confusion between CSA and TSA
Students assume that the curved surface area (CSA) and the total surface area (TSA) of a frustum are the same. This confusion arises because both involve the slant height and the radii of the bases. Always remember that CSA involves only the curved side of the frustum, while TSA includes the curved surface and both bases.
Mistake 1
Using height instead of slant height
Some students mistakenly use the vertical height (h) instead of the slant height (l) when finding the curved surface area. Remember the formula for curved surface area is π(R + r)l, so always use the slant height.
Mistake 2
Using the wrong value for π
A common mistake of students is using 22 as the value of π. Instead, use accurate values like 22/7 or 3.14. π is a special number in math, and using the correct value is important for getting the right answer.
Mistake 3
Forgetting to include both base areas in the total surface area
Students often calculate only the curved surface and forget to add the areas of both circular bases. Always include all parts when calculating the total surface area. Total surface area = curved surface area + area of top base + area of bottom base.
Mistake 4
Assuming curved surface area and lateral surface area are different
Some students mistakenly believe that curved surface area and lateral surface area are different, but for a frustum, they mean the same thing. So, use the same formula for both.
Mistake 5
Solved Examples of Surface Area of Frustum of Cone
Find the curved surface area of a frustum with top radius 5 cm, bottom radius 8 cm, and slant height 10 cm.
CSA = 408 cm²
Problem 1
Given R = 5 cm, r = 8 cm, l = 10 cm. Use the formula: CSA = π(R + r)l = (22/7) × (5 + 8) × 10 = (22/7) × 13 × 10 = 22 × 130/7 = 408 cm²
Find the total surface area of a frustum with top radius 4 cm, bottom radius 7 cm, and slant height 12 cm.
Explanation
TSA = 692.2 cm²
Problem 2
Use the formula: TSA = π(R² + r² + (R + r)l) = 3.14 × (4² + 7² + (4 + 7) × 12) = 3.14 × (16 + 49 + 11 × 12) = 3.14 × (65 + 132) = 3.14 × 197 = 692.2 cm²
A frustum has a top radius of 6 cm, bottom radius of 10 cm, and a height of 15 cm. Find the total surface area.
Explanation
TSA = 1222.56 cm²
Problem 3
Find the slant height using: l = √((10 - 6)² + 15²) = √(16 + 225) = √241 = 15.52 cm Use the TSA formula: TSA = π(R² + r² + (R + r)l) = 3.14 × (6² + 10² + (6 + 10) × 15.52) = 3.14 × (36 + 100 + 248.32) = 3.14 × 384.32 = 1222.56 cm²
Find the curved surface area of a frustum with top radius 3 cm, bottom radius 6 cm, and slant height 8 cm.
Explanation
CSA = 226.08 cm²
Problem 4
CSA = π(R + r)l = (22/7) × (3 + 6) × 8 = (22/7) × 9 × 8 = 22 × 72/7 = 226.08 cm²
The slant height of a frustum is 9 cm, and its curved surface area is 396 cm². The top radius is 7 cm. Find the bottom radius.
Explanation
Bottom Radius = 9 cm
It is the total area that covers the outer surface of the frustum, including its curved side and the areas of both circular bases.
1.What are the two types of surface area in a frustum?
2.What is the difference between slant height and height?
3.Is curved surface area the same as lateral surface area?
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.
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.
Fun Fact
: She has songs for each table which helps her to remember the tables
