Surface Area of Right Cone
2026-02-28 23:40 Diff
https://brightchamps.com/en-us/math/measurement/surface-area-of-right-cone

183 Learners

Last updated on August 5, 2025

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.

Explore Our Programs

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.

Problem 1

Given r = 8 cm, l = 12 cm. Use the formula: CSA = πrl = (3.14) × 8 × 12 = 301.44 cm²

Okay, lets begin

Find the total surface area of a right cone with radius 4 cm and slant height 15 cm.

Explanation

TSA = 238.64 cm²

Well explained 👍

Problem 2

Use the formula: TSA = πr(r + l) = 3.14 × 4 (4 + 15) = 3.14 × 4 × 19 = 238.64 cm²

Okay, lets begin

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

Well explained 👍

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

Okay, lets begin

Find the curved surface area of a right cone with radius 5 cm and slant height 7 cm.

Explanation

CSA = 110 cm²

Well explained 👍

Problem 4

CSA = πrl = (22/7) × 5 × 7 = 22 × 5 = 110 cm²

Okay, lets begin

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

Well explained 👍

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?

Curved surface area and total surface area are the two types of surface area in a right cone.

2.What is the difference between slant height and height?

Slant height is the length from the tip to the edge of the base. Height is the straight line from the tip to the center of the base.

3.Is curved surface area the same as lateral surface area?

Yes, in cones, both curved and lateral surface area mean the same.

4.What unit is surface area measured in?

Surface area is always measured in square units like cm², m², or in².

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.

What Is Measurement? 📏 | Easy Tricks, Units & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math

▶

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