101 LearnersLast updated on August 5th, 2025
Surface Area of Right Circular Cone
A right circular cone is a 3-dimensional shape with a circular base and a pointed vertex directly above the center of the base. The surface area of a right circular cone is the total area covered by its outer surface, including both its curved surface and its base. In this article, we will explore the surface area of a right circular cone.
What is the Surface Area of a Right Circular Cone?
The surface area of a right circular cone is the total area occupied by the boundary or surface of the cone. It is measured in square units.
A right circular cone is a 3D shape formed by rotating a right triangle around one of its sides. It has a circular base at the bottom and a sharp point at the top called the vertex.
A right circular cone has a curved surface and a flat base, so it has two surface areas: the curved surface area and the total surface area.
Surface Area of a Right Circular Cone Formula
A right circular cone has a curved surface, which includes two types of surface areas: the curved surface area and the total surface area.
Look at the right circular cone below to see its surface area, height (h), slant height (l), and radius (r).
A right circular cone has two types of surface areas:
Curved Surface Area of a Right Circular Cone
Total Surface Area of a Right Circular Cone
Curved Surface Area of a Right Circular Cone
The area of the curved part of the cone, excluding its base, is known as the curved surface area of a right circular cone. The curved surface area of the cone is also called the lateral surface area.
The formula for the CSA (Curved Surface Area) of a right circular cone is given as:
Curved Surface Area = 𝜋rl square units
Here, r is the radius of the base of the cone. l is the slant height of the cone.
Total Surface Area of a Right Circular Cone
The total area occupied by the right circular cone, including the area of the curved surface and the area of the circular base, is known as the total surface area.
The total surface area is calculated using the formula: Total surface area = 𝜋r(r + l) square units Where r is the radius of the base of the cone. l is the slant height of the cone. Derivation of the Total Surface Area of a Right Circular Cone
To find the total surface area, imagine slicing it from the tip to the edge of the base and unfolding it. When spread out flat, the curved surface becomes a part of a circle, called a sector.
Consider a right circular cone with height (h), radius of the base (r), and slant height (l). Total surface area = base area + curved surface area
Here, the base area = 𝜋r² Curved surface area = 𝜋rl
Substituting the formulas into the total surface area,
Total surface area, T = 𝜋r² + 𝜋rl
Take common terms out: T = 𝜋r(r + l)
Therefore, the total surface area, T = 𝜋r(r + l)
Volume of a Right Circular Cone
The volume of a right circular cone shows how much space is inside it. It tells us the capacity of the cone. The volume is one-third of the volume of a cylinder with the same height and base. The volume can be found using the formula: Volume = ⅓(𝜋r²h) cubic units
Confusion between CSA and TSA
Students assume that the curved surface area (CSA) and the total surface area (TSA) of a cone are the same. This confusion arises because both involve the slant height and the radius.
Always remember that CSA is only for the curved side of the cone, whereas TSA includes the curved surface and the base.
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 𝜋rl, so always use the slant height.
Mistake 2
Using the wrong value for 𝜋
A common mistake 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 the base area in the total surface area
Students often calculate only the curved surface and forget to add the circular base. Always include both parts when calculating the total surface area. Total surface area = curved surface area + base area.
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 in a cone, they mean the same thing. So, use the same formula for both.
Mistake 5
Solved Examples of Surface Area of Right Circular Cone
Find the curved surface area of a cone with a radius of 4 cm and a slant height of 9 cm.
CSA = 113.04 cm²
Problem 1
Given r = 4 cm, l = 9 cm. Use the formula: CSA = 𝜋rl = 3.14 × 4 × 9 = 113.04 cm²
Find the total surface area of a cone with a radius of 6 cm and a slant height of 10 cm.
Explanation
TSA = 301.44 cm²
Problem 2
Use the formula: TSA = 𝜋r(r + l) = 3.14 × 6 (6 + 10) = 3.14 × 6 × 16 = 3.14 × 96 = 301.44 cm²
A cone has a radius of 8 cm and a height of 5 cm. Find the total surface area.
Explanation
TSA = 284.48 cm²
Problem 3
Find the slant height using: l = √(r² + h²) = √(8² + 5²) = √(64 + 25) = √89 ≈ 9.43 cm Use the TSA formula: TSA = 𝜋r(r + l) = 3.14 × 8 × (8 + 9.43) = 3.14 × 8 × 17.43 = 3.14 × 139.44 = 438.41 cm²
Find the curved surface area of a cone with a radius of 7 cm and a slant height of 12 cm.
Explanation
CSA = 263.76 cm²
Problem 4
CSA = 𝜋rl = 3.14 × 7 × 12 = 263.76 cm²
The slant height of a cone is 20 cm, and its curved surface area is 440 cm². Find the radius.
Explanation
Radius = 7 cm
It is the total area that covers the outside of the cone, including its curved side and the base.
1.What are the two types of surface area in a right circular 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 Right Circular Cone
Students often make mistakes while calculating the surface area of a right circular cone, leading to incorrect answers. Below are some common mistakes and 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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