Area of Cone
2026-02-28 08:42 Diff
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Last updated on September 13, 2025

Area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of a cone.

What is the Area of a Cone?

A cone is a three-dimensional shape that tapers smoothly from a flat circular base to a point called the apex or vertex. The surface area of a cone is the total area of its lateral (curved) surface and its base.

Area of the Cone Formula

To find the surface area of a cone, we use the formula: Lateral Area = π × r × l, where r is the radius of the base and l is the slant height.

The total surface area is given by: Total Surface Area = π × r × l + π × r². Now let’s see how the formula is derived.

Derivation of the formula: The lateral surface area is calculated by unwrapping the cone into a sector of a circle, which is part of a larger circle with a radius equal to the slant height l. The arc length of the sector is equal to the circumference of the base of the cone, i.e., 2πr.

Thus, the lateral surface area of the cone = π × r × l. Adding the area of the base, which is a circle of radius r, gives us the total surface area: Total Surface Area = π × r × l + π × r².

How to Find the Area of a Cone?

We can find the area of a cone using the following method:

Method Using Radius and Slant Height

If the radius r and slant height l are given, we find the surface area of the cone using the formulas: Lateral Area = π × r × l and Total Surface Area = π × r × l + π × r². For example, if r is 3 cm and l is 5 cm.

What will be the total surface area of the cone? Total Surface Area = π × r × l + π × r² = π × 3 × 5 + π × 3² = 15π + 9π = 24π cm² The total surface area of the cone is approximately 75.36 cm² (using π ≈ 3.14).

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Unit of Area of Cone

We measure the area of a cone in square units. The measurement depends on the system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²). In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

Special Cases or Variations for the Area of Cone

Since a cone has a circular base and a curved surface, there are specific cases for calculating the area:

Case 1: Surface Area Using Radius and Slant Height If the radius and slant height are given, use the formula Total Surface Area = π × r × l + π × r².

Case 2: Use of Height If the height h of the cone is given instead of the slant height, use the Pythagorean theorem to find the slant height: l = √(r² + h²).

Tips and Tricks for Area of Cone

To ensure correct results while calculating the area of a cone, here are some tips and tricks: The slant height is different from the perpendicular height of the cone.

The total surface area includes both the lateral surface area and the base area. Use the Pythagorean theorem to find the slant height if only the height and radius are given.

Common Mistakes and How to Avoid Them in Area of Cone

It is common to make mistakes while finding the area of a cone. Let’s take a look at some common mistakes.

Problem 1

The radius of a cone's base is 6 m and the slant height is 10 m. What will be the total surface area?

Okay, lets begin

We will find the total surface area as approximately 301.44 m².

Explanation

Here, the radius r is 6 m and the slant height l is 10 m.

The total surface area = π × r × l + π × r² = π × 6 × 10 + π × 6² = 60π + 36π = 96π ≈ 301.44 m².

Well explained 👍

Problem 2

What will be the total surface area of the cone if the radius is 4 cm and the slant height is 9 cm?

Okay, lets begin

We will find the total surface area as approximately 163.28 cm².

Explanation

Using the formula

Total Surface Area = π × r × l + π × r², with r = 4 cm and l = 9 cm, we get:

Total Surface Area = π × 4 × 9 + π × 4² = 36π + 16π = 52π ≈ 163.28 cm².

Well explained 👍

Problem 3

The total surface area of a cone is 282.6 cm² and the radius of the base is 6 cm. Find the slant height.

Okay, lets begin

We find the slant height as approximately 8 cm.

Explanation

Using the formula:

Total Surface Area = π × r × l + π × r², substitute the known values to find the slant height l: 282.6 = π × 6 × l + π × 6².

Solving for l gives us l = (282.6 - 36π) / (6π) ≈ 8 cm.

Well explained 👍

Problem 4

Find the total surface area of the cone if its radius is 7 cm and the perpendicular height is 24 cm.

Okay, lets begin

We will find the total surface area as approximately 791.28 cm².

Explanation

First, find the slant height using the Pythagorean theorem: l = √(r² + h²) = √(7² + 24²) = 25 cm.

Then use the formula Total Surface Area = π × r × l + π × r² = π × 7 × 25 + π × 7² = 175π + 49π = 224π ≈ 791.28 cm².

Well explained 👍

Problem 5

Help Jane find the total surface area of the cone if the base radius is 5 m and the slant height is 12 m.

Okay, lets begin

We will find the total surface area as approximately 267.94 m².

Explanation

The base radius is 5 m and the slant height is 12 m.

Using the formula Total Surface Area = π × r × l + π × r², we have:

Total Surface Area = π × 5 × 12 + π × 5² = 60π + 25π = 85π ≈ 267.94 m².

Well explained 👍

FAQs on Area of Cone

1.Is it possible for the area of a cone to be negative?

No, the area of a cone can never be negative. The area of any shape will always be positive.

2.How to find the surface area of a cone if the radius and slant height are given?

If the radius and slant height are given, then we find the surface area using the formula: Total Surface Area = π × r × l + π × r².

3.How to find the surface area of a cone if only the height and radius are given?

If the height and radius are given, find the slant height using l = √(r² + h²), then use the formula Total Surface Area = π × r × l + π × r².

4.What is meant by the surface area of a cone?

The surface area of a cone is the total area of its lateral (curved) surface plus the area of its base.

5.How is the volume of the cone calculated?

The volume of a cone is calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the perpendicular height.

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