Surface Area of Spherical Shell
2026-02-28 17:52 Diff
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Last updated on August 16, 2025

A spherical shell is a 3-dimensional shape that consists of two concentric spheres with different radii. The surface area of a spherical shell refers to the total area covered by its outer surface. In this article, we will learn about the surface area of a spherical shell.

What is the Surface Area of a Spherical Shell?

The surface area of a spherical shell is the difference in the surface areas of the outer and inner spheres. It is measured in square units. A spherical shell is formed by two concentric spheres where the outer sphere has a larger radius than the inner sphere. The surface area is calculated by subtracting the surface area of the inner sphere from the surface area of the outer sphere.

Surface Area of a Spherical Shell Formula

A spherical shell has only one surface area type, which is the total surface area of the shell itself. Consider the radii of the outer sphere (R) and the inner sphere (r). The formula for the surface area of a spherical shell is: Surface Area of Spherical Shell = 4π(R² - r²)

Calculation of Surface Area of a Spherical Shell

The surface area of a spherical shell is calculated by finding the difference between the outer and inner spheres' surface areas. The formula for the surface area of a sphere is 4πr². Therefore, the surface area of a spherical shell is: Surface Area = 4πR² - 4πr² = 4π(R² - r²) Here, R is the radius of the outer sphere, and r is the radius of the inner sphere.

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Volume of a Spherical Shell

The volume of a spherical shell shows how much space is enclosed between the two concentric spheres. The volume is calculated using the formula: Volume = 4/3π(R³ - r³) (cubic units)

Confusion between Surface Areas of Spheres and Spherical Shells

Students may confuse the formulas for the surface area of a single sphere with that of a spherical shell. Remember, the surface area of a spherical shell involves the difference in surface areas of two spheres.

Problem 1

Given R = 10 cm, r = 7 cm. Use the formula: Surface Area = 4π(R² - r²) = 4π(10² - 7²) = 4π(100 - 49) = 4π(51) = 4 × 3.14 × 51 = 1,318.4 cm²

Okay, lets begin

Calculate the surface area of a spherical shell with an outer radius of 15 cm and an inner radius of 12 cm.

Explanation

Surface area = 1,696.5 cm²

Well explained 👍

Problem 2

Use the formula: Surface Area = 4π(R² - r²) = 4 × 3.14 × (15² - 12²) = 4 × 3.14 × (225 - 144) = 4 × 3.14 × 81 = 1,696.5 cm²

Okay, lets begin

A spherical shell has an outer radius of 8 cm and an inner radius of 5 cm. Find the surface area.

Explanation

Surface area = 615.44 cm²

Well explained 👍

Problem 3

Use the formula: Surface Area = 4π(R² - r²) = 4 × 3.14 × (8² - 5²) = 4 × 3.14 × (64 - 25) = 4 × 3.14 × 39 = 615.44 cm²

Okay, lets begin

Determine the surface area of a spherical shell with an outer radius of 9 cm and an inner radius of 6 cm.

Explanation

Surface area = 848.23 cm²

Well explained 👍

Problem 4

Use the formula: Surface Area = 4π(R² - r²) = 4 × 3.14 × (9² - 6²) = 4 × 3.14 × (81 - 36) = 4 × 3.14 × 45 = 848.23 cm²

Okay, lets begin

The surface area of a spherical shell is 2,513.28 cm², and the outer radius is 14 cm. Find the inner radius.

Explanation

Inner radius = 10 cm

Well explained 👍

It is the total area that covers the outer surface of a spherical shell, calculated as the difference between the surface areas of the outer and inner spheres.

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

The formula is 4π(R² - r²), where R is the outer radius and r is the inner radius.

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

A sphere is a solid shape with all points equidistant from the center, while a spherical shell is hollow, formed by two concentric spheres.

3.What unit is surface area measured in?

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

4.Can the surface area of a spherical shell be negative?

No, the surface area cannot be negative. Ensure that the outer radius is larger than the inner radius.

Common Mistakes and How to Avoid Them in the Surface Area of a Spherical Shell

Students often make mistakes while calculating the surface area of a spherical shell, which leads 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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: She has songs for each table which helps her to remember the tables