101 LearnersLast updated on July 31st, 2025
Surface Area of Right Cone
A right cone is a 3-dimensional shape with a circular base. The surface area of a right cone is the total area covered by its outer surface. The surface area includes both its curved surface and its base, while the curved surface area only includes the slanted part of the cone. In this article, we will learn about the surface area of a right cone.
What is the Surface Area of a Right Cone?
The surface area of a right cone is the total area occupied by the boundary or surface of the cone. It is measured in square units.
A right cone is a 3D shape formed by rotating a right triangle around one of its legs. It has a circular base at the bottom and a sharp point at the top called the vertex.
A right cone has a curved side and a flat base, so it has two surface areas: the curved surface area and the total surface area. Right cones are a type of cone where the tip of the cone is directly above the center of the base, forming a neat, symmetrical shape.
Surface Area of a Right Cone Formula
A right cone has a curved surface, and it has two types of surface areas: the curved surface area and the total surface area.
Look at the right cone below to see its surface area, height (h), slant height (l), and radius (r). A right cone has two types of surface areas: Curved Surface Area of a Right Cone Total Surface Area of a Right Cone
Curved Surface Area of a Right Cone
The area of the curved part of the right cone, excluding its base, is known as the curved surface area of a right cone. The curved surface area of the cone is also called the lateral surface area.
The formula for the CSA (Curved Surface Area) of the right cone is given as: Curved Surface Area = πrl square units Here, r is the radius of the base of the right cone. l is the slant height of the right cone.
Total Surface Area of a Right Cone
The total area occupied by the right cone, including the area of the curved surface and the area of the circular base, is known as the total surface area of the right cone.
The total surface area of a cone is calculated by using the formula: Total Surface Area = πr(r + l) square units Where r is the radius of the base of the right cone. l is the slant height of the right cone.
Derivation of the Total Surface Area of a Right Cone
To find the total surface area of a right cone, imagine slicing it from the tip to the edge of the base and unfolding it. When we spread it out flat, the curved surface becomes a part of a circle, called a sector.
Let us take a right cone with height (h), radius of the base (r), and a slant height (l).
Total surface area of a right cone = base area of a cone + curved surface area of a cone
Here, the base area of a cone = πr² Curved surface area of a cone = πrl
Substituting the formulas into the total surface area, Total surface area of a cone, T = πr² + πrl
Take the common terms out: T = πr(r + l)
Therefore, the total surface area of a right cone T = πr(r + l)
Volume of a Right Cone
The volume of a right cone shows how much space is inside it. It tells us how much space is inside the cone or how much it can hold. It is one-third of the volume of a cylinder with the same height and base.
The volume of a right cone can be found by 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 right cone are the same. This confusion arises because both involve the slant height and the radius. Always remember that CSA is used for only the curved side of the cone, and 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 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 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 Cone
Find the curved surface area of a right cone with a radius of 8 cm and a slant height of 12 cm.
CSA = 301.44 cm²
Problem 1
Given r = 8 cm, l = 12 cm. Use the formula: CSA = πrl = (3.14) × 8 × 12 = 301.44 cm²
Find the total surface area of a right cone with radius 4 cm and slant height 15 cm.
Explanation
TSA = 238.64 cm²
Problem 2
Use the formula: TSA = πr(r + l) = 3.14 × 4 (4 + 15) = 3.14 × 4 × 19 = 238.64 cm²
A right cone has a radius of 10 cm and a height of 24 cm. Find the total surface area.
Explanation
TSA = 1130.4 cm²
Problem 3
Find the slant height using: l = √(r² + h²) = √(10² + 24²) = √(100 + 576) = √676 = 26 cm Use the TSA formula: TSA = πr(r + l) = 3.14 × 10 × (10 + 26) = 3.14 × 10 × 36 = 1130.4 cm²
Find the curved surface area of a right cone with radius 5 cm and slant height 7 cm.
Explanation
CSA = 110 cm²
Problem 4
CSA = πrl = (22/7) × 5 × 7 = 22 × 5 = 110 cm²
The slant height of a right cone is 18 cm, and its curved surface area is 792 cm². Find the radius.
Explanation
Radius = 14 cm
It is the total area that covers the outside of the right cone, including its curved side and the base.
1.What are the two types of surface area in a right 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 Cone
Students often make mistakes while calculating the surface area of a right cone, which leads to wrong 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
