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130 LearnersLast updated on August 30, 2025
Surface Area of Spherical Cap
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.
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.
Mistake 1
Incorrect Use of Radius and Height
Some students mistakenly use the diameter of the sphere instead of the radius (R) when calculating the surface area. Always ensure you use the radius, and remember that it is half of the diameter. Also, ensure the height (h) used is the height of the cap, not the sphere.
Mistake 2
Using the Wrong Value for π
A common mistake of students is using 22 as the value of π. Instead, use accurate values like 22/7 or 3.14. π is a special number in math, and using the correct value is important for getting the right answer.
Mistake 3
Forgetting the Height in the Surface Area Calculation
Students often forget to incorporate the height (h) in the surface area formula for a spherical cap. The height is crucial as it determines the extent of the cap on the sphere.
Mistake 4
Solved Examples of Surface Area of Spherical Cap
Find the surface area of a spherical cap with a sphere radius of 10 cm and a cap height of 4 cm.
Surface Area = 251.2 cm²
Problem 1
Given R = 10 cm, h = 4 cm. Use the formula: Surface Area = 2πRh = 2 × 3.14 × 10 × 4 = 251.2 cm²
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
Problem 2
Use the formula: Surface Area = 2πRh = 2 × 3.14 × 7 × 3 = 131.88 cm²
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
Problem 3
Use the formula: Surface Area = 2πRh = 2 × 3.14 × 12 × 5 = 376.8 cm²
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
Problem 4
Surface Area = 2πRh = 2 × 3.14 × 15 × 6 = 565.2 cm²
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
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?
2.What is the difference between a sphere and a spherical cap?
3.Is the height the same as the radius in a spherical cap?
4.What unit is surface area measured in?
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.
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
