110 LearnersLast updated on August 5th, 2025
Surface Area of Half a Sphere
A half sphere, or hemisphere, is a 3-dimensional shape that is half of a full sphere. The surface area of a hemisphere includes both its curved surface and its flat circular base. In this article, we will learn about the surface area of a half sphere.
What is the Surface Area of a Half Sphere?
The surface area of a half sphere is the total area occupied by the outer surface of a hemisphere. It is measured in square units.
A hemisphere is formed by slicing a sphere into two equal halves, resulting in a curved surface and a flat circular base.
The surface area of a hemisphere includes both the curved surface area and the total surface area.
Surface Area of a Half Sphere Formula
A half sphere has a curved surface and a flat circular base. Look at the hemisphere below to see its surface area and radius(r).
A half sphere has two types of surface areas:
Curved Surface Area of a Half Sphere
Total Surface Area of a Half Sphere
Curved Surface Area of a Half Sphere
The area of the curved part of the hemisphere, excluding its base, is known as the curved surface area of a half sphere.
The formula for the CSA (Curved Surface Area) of a hemisphere is given as: Curved Surface Area = 2𝜋r² square units
Here, r is the radius of the base of the hemisphere.
Total Surface Area of a Half Sphere
The total area occupied by the half sphere, including the area of the curved surface and the area of the circular base, is known as the total surface area of a half sphere.
The total surface area of a half sphere is calculated using the formula: Total surface area = 3𝜋r² square units Where r is the radius of the base of the hemisphere.
The total surface area of a half sphere = base area of a half sphere + curved surface area of a half sphere.
Here, the base area of a half sphere = 𝜋r²
Curved surface area of a half sphere = 2𝜋r²
Substituting the formulas into the total surface area,
Total surface area of a half sphere, T = 𝜋r² + 2𝜋r²
Therefore, the total surface area of a half sphere T = 3𝜋r²
Volume of a Half Sphere
The volume of a half sphere shows how much space is inside it. It tells us how much space the half sphere can hold. The volume of a half sphere can be found using the formula: Volume = (2/3)𝜋r³ (cubic units)
Confusion between CSA and TSA
Students assume that the curved surface area (CSA) and the total surface area (TSA) of a half sphere are the same. This confusion arises because both involve the radius. Always remember that CSA covers only the curved side of the half sphere, and TSA includes the curved surface and the base.
Mistake 1
Using the wrong value for 𝜋
A common mistake is using incorrect values for 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 2
Forgetting to include the base area in the total surface area
Students often calculate only the curved surface and forget to add the circular base. Always include both parts when calculating the total surface area. Total surface area = curved surface area + base area.
Mistake 3
Assuming curved surface area and lateral surface area are different
Some students mistakenly believe that curved surface area and lateral surface area are different, but in a hemisphere, they mean the same thing. So, use the same formula for both.
Mistake 4
Solved Examples of Surface Area of Half a Sphere
Find the curved surface area of a half sphere with a radius of 7 cm.
CSA = 308 cm²
Problem 1
Given r = 7 cm. Use the formula: CSA = 2𝜋r² = 2 × (22/7) × 7 × 7 = 2 × 22 × 7 = 308 cm²
Find the total surface area of a half sphere with a radius of 5 cm.
Explanation
TSA = 235.5 cm²
Problem 2
Use the formula: TSA = 3𝜋r² = 3 × 3.14 × 5 × 5 = 3 × 3.14 × 25 = 3 × 78.5 = 235.5 cm²
A half sphere has a radius of 6 cm. Find the total surface area.
Explanation
TSA = 339.12 cm²
Problem 3
Use the TSA formula: TSA = 3𝜋r² = 3 × 3.14 × 6 × 6 = 3 × 3.14 × 36 = 3 × 113.04 = 339.12 cm²
Find the curved surface area of a half sphere with a radius of 3.5 cm.
Explanation
CSA = 77 cm²
Problem 4
CSA = 2𝜋r² = 2 × (22/7) × 3.5 × 3.5 = 2 × 22 × 0.5 × 3.5 = 22 × 3.5 = 77 cm²
The curved surface area of a half sphere is 660 cm². Find the radius.
Explanation
Radius = 10.5 cm
It is the total area that covers the outside of the half sphere, including its curved side and the base.
1.What are the two types of surface area in a half sphere?
2.What is the difference between curved surface area and total surface area?
3.Is curved surface area the same as lateral surface area?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of a Half Sphere
Students often make mistakes while calculating the surface area of a half sphere, leading to wrong answers. Below are some common mistakes and the ways to avoid them.
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Seyed Ali Fathima S
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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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