Surface Area of Spherical Cap
2026-02-28 08:44 Diff
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Last updated on August 30, 2025

A spherical cap is a portion of a sphere that is cut off by a plane. The surface area of the spherical cap is the total area covered by its outer surface. In this article, we will learn about the surface area of a spherical cap.

What is the Surface Area of a Spherical Cap?

The surface area of a spherical cap is the total area occupied by the boundary or surface of a spherical cap.

It is measured in square units. A spherical cap is formed by slicing a sphere with a plane, resulting in a dome-like shape.

The cap has a curved surface, and its area depends on the radius of the sphere and the height of the cap.

Surface Area of a Spherical Cap Formula

The surface area of a spherical cap is calculated using the formula that involves the radius of the sphere (R) and the height of the cap (h). The formula for the surface area of a spherical cap is:

Surface Area = 2πRh

Here, R is the radius of the sphere, and h is the height of the cap.

Derivation of the Surface Area of a Spherical Cap

To derive the surface area of a spherical cap, consider a sphere of radius R.

If a plane slices the sphere, creating a cap with height h, the surface area of the cap can be calculated using the formula:

Surface Area = 2πRh

This formula results from integrating the curved surface area of the sphere over the height of the cap.

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Volume of a Spherical Cap

The volume of a spherical cap tells us how much space is enclosed within the cap.

It is calculated by the formula: Volume = (1/3)πh²(3R - h) where R is the radius of the sphere, and h is the height of the cap.

Confusion between Sphere and Spherical Cap Formulas

Students sometimes confuse the formulas for the surface area of a full sphere and the surface area of a spherical cap. Remember that the surface area of a sphere is 4πR², while the surface area of a spherical cap is 2πRh.

Problem 1

Given R = 10 cm, h = 4 cm. Use the formula: Surface Area = 2πRh = 2 × 3.14 × 10 × 4 = 251.2 cm²

Okay, lets begin

Calculate the surface area of a spherical cap on a sphere with a radius of 7 cm and a cap height of 3 cm.

Explanation

Surface Area = 131.88 cm²

Well explained 👍

Problem 2

Use the formula: Surface Area = 2πRh = 2 × 3.14 × 7 × 3 = 131.88 cm²

Okay, lets begin

A spherical cap has a height of 5 cm on a sphere with a radius of 12 cm. Find the surface area.

Explanation

Surface Area = 376.8 cm²

Well explained 👍

Problem 3

Use the formula: Surface Area = 2πRh = 2 × 3.14 × 12 × 5 = 376.8 cm²

Okay, lets begin

Find the surface area of a spherical cap with a radius of 15 cm and a height of 6 cm.

Explanation

Surface Area = 565.2 cm²

Well explained 👍

Problem 4

Surface Area = 2πRh = 2 × 3.14 × 15 × 6 = 565.2 cm²

Okay, lets begin

The surface area of a spherical cap is 188.4 cm², and the radius of the sphere is 9 cm. Find the height of the cap.

Explanation

Height = 3.33 cm

Well explained 👍

It is the total area that covers the curved surface of the cap section of a sphere.

1.What is the formula for the surface area of a spherical cap?

The formula is 2πRh, where R is the radius of the sphere and h is the height of the cap.

2.What is the difference between a sphere and a spherical cap?

A sphere is a full 3D round object, while a spherical cap is a portion of a sphere cut by a plane.

3.Is the height the same as the radius in a spherical cap?

No, the height (h) is the distance from the base of the cap to the top, whereas the radius (R) is the radius of the sphere.

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 Spherical Cap

Students often make mistakes while calculating the surface area of a spherical cap, which leads to wrong answers. Below are some common mistakes and the 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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: She has songs for each table which helps her to remember the tables