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104 LearnersLast updated on September 4, 2025
Surface Area of Truncated Cone
A truncated cone, or frustum, is a 3-dimensional shape that resembles a cone with its top cut off parallel to the base. The surface area of a truncated cone is the total area covered by its outer surface, including both its curved surface and its two circular bases. In this article, we will learn about the surface area of a truncated cone.
What is the Surface Area of a Truncated Cone?
The surface area of a truncated cone is the total area occupied by the boundary or surface of the truncated cone. It is measured in square units.
A truncated cone is formed by slicing the top off a cone parallel to its base. This shape has two circular bases: a larger base and a smaller base. It also has a curved surface connecting the two bases.
Therefore, the surface area includes the areas of these two bases and the curved surface area. The truncated cone can have varying slant heights depending on the angle of the cut.
Surface Area of a Truncated Cone Formula
A truncated cone has a curved surface and two bases, each with its area. The surface area (SA) of a truncated cone is calculated using the formula:
SA = pi (r_1 + r_2) x l + pi r_12 + pi r_22 where (r_1) and (r_2) are the radii of the two bases, and l is the slant height of the truncated cone.
Curved Surface Area of a Truncated Cone
The curved surface area of a truncated cone is the area of the slanted portion that connects the two circular bases.
The formula for the curved surface area (CSA) is: CSA = pi (r_1 + r_2) x l where (r_1) and (r_2) are the radii of the two bases, and l is the slant height of the truncated cone.
Total Surface Area of a Truncated Cone
The total surface area of a truncated cone includes the areas of both circular bases and the curved surface.
The formula is: Total Surface Area = pi (r_1 + r_2) x l + \pi r_12 + pi r_22 where (r_1) and (r_2) are the radii of the two bases, and l is the slant height.
To derive this formula, imagine unrolling the curved surface into a sector of a larger circle. The radius of this circle is the slant height, and the arc length is the difference in the circumferences of the two bases.
Volume of a Truncated Cone
The volume of a truncated cone shows how much space is inside it. It is one-third of the height times the sum of the areas of the two bases plus the square root of the product of the bases.
The formula is: Volume = 1/3 pi h (r_12 + r_22 + r_1 x r_2) where h is the height of the truncated cone, and (r_1) and (r_2) are the radii of the two bases.
Confusion between Curved Surface Area and Total Surface Area
Students sometimes confuse the curved surface area (CSA) with the total surface area (TSA). Remember, CSA includes only the slanted surface, while TSA includes both bases and the slanted surface.
Mistake 1
Using Height Instead of Slant Height
Some students mistakenly use the vertical height \(h\) instead of the slant height \(l\) when calculating the curved surface area. Always use the slant height in the formula \(\pi (r_1 + r_2) \times l\).
Mistake 2
Using the Wrong Value for π
A common mistake is using an incorrect value for π. Always use 3.14 or 22/7 for accurate results.
Mistake 3
Forgetting to Include Both Bases in Total Surface Area
Students often calculate only the curved surface and forget to add the areas of both circular bases. Ensure to include both parts when calculating the total surface area.
Mistake 4
Assuming the Two Bases Are the Same Size
Some students mistakenly assume the two bases of a truncated cone are the same size. Remember, the radii \(r_1\) and \(r_2\) can be different, affecting both the total and curved surface areas.
Mistake 5
Solved Examples of Surface Area of Truncated Cone
Find the curved surface area of a truncated cone with radii 5 cm and 3 cm, and a slant height of 7 cm.
CSA = 176 cm²
Problem 1
Given (r_1 = 5) cm, (r_2 = 3) cm, (l = 7) cm. Use the formula: CSA = pi (r_1 + r_2) x l = (22/7) x (5 + 3) x 7 = (22/7) x 8 x 7 = 176 cm²
Find the total surface area of a truncated cone with radii 6 cm and 4 cm, and a slant height of 10 cm.
Explanation
TSA = 376.8 cm²
Problem 2
Use the formula: TSA = pi (r_1 + r_2) x l + pi r_1² + \pi r_2² = 3.14 x (6 + 4) x 10 + 3.14 x 6² + 3.14 x 4² = 3.14 x 10 x 10 + 3.14 x 36 + 3.14 x 16 = 314 + 113.04 + 50.24 = 376.8 cm²
A truncated cone has a larger base radius of 7 cm, a smaller base radius of 5 cm, and a slant height of 12 cm. Find the total surface area.
Explanation
TSA = 577.76 cm²
Problem 3
Use the formula: TSA = pi (r_1 + r_2) x l + pi r_1² + pi r_2² = 3.14 x (7 + 5) x 12 + 3.14 x 7² + 3.14 x 5² = 3.14 x12 x 12 + 3.14 x 49 + 3.14 x 25= 452.16 + 153.86 + 78.5= 577.76 cm²
Find the curved surface area of a truncated cone with radii 4.5 cm and 2.5 cm, and a slant height of 8 cm.
Explanation
CSA = 176 cm²
Problem 4
CSA = pi (r_1 + r_2) x l= (22/7) x (4.5 + 2.5) x 8 = (22/7) x 7 x 8 = 176 cm²
The slant height of a truncated cone is 15 cm, and its curved surface area is 660 cm². Find the sum of the radii of the bases.
Explanation
Sum of radii = 14 cm
It is the total area that covers the outside of the truncated cone, including its curved side and the two bases.
1.What are the two types of surface areas in a truncated cone?
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 Truncated Cone
Students often make mistakes while calculating the surface area of a truncated cone, leading to incorrect answers. Below are some common mistakes and ways to avoid them.
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Seyed Ali Fathima S
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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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