104 LearnersLast updated on August 16th, 2025
Surface Area of Spherical Shell
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.
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.
Mistake 1
Using Radii Incorrectly
Some students mistakenly use the inner radius when they should be using the difference between the squared radii. Always ensure you are using both radii correctly in the formula.
Mistake 2
Using the Wrong Value for π
A common mistake is using an incorrect approximation of π. Use accurate values like 22/7 or 3.14 to ensure precision in calculations.
Mistake 3
Forgetting the Squaring of Radii
Students often forget to square the radii in the formula. Always square the radii as required in the formula for surface area.
Mistake 4
Misinterpreting the Formula
Some students might misinterpret the formula and perform operations incorrectly. Make sure to follow the order: find the difference of squared radii first, then multiply by 4π.
Mistake 5
Solved Examples of Surface Area of Spherical Shell
Find the surface area of a spherical shell with an outer radius of 10 cm and an inner radius of 7 cm.
Surface area = 1,318.4 cm²
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
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
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
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
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
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
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
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
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?
2.What is the difference between a sphere and a spherical shell?
3.What unit is surface area measured in?
4.Can the surface area of a spherical shell be negative?
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.
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
