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Last updated on September 15, 2025

Area of Hemisphere

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Area is the space inside the boundaries of a three-dimensional object. There are different formulas for finding the surface area of various shapes/objects. These are widely used in architecture and design. In this section, we will find the area of a hemisphere.

Area of Hemisphere for US Students
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What is the Area of a Hemisphere?

A hemisphere is a three-dimensional shape that represents half of a sphere. It has a curved surface area and a flat circular base.

 

The area of a hemisphere is the total surface area, which includes both the curved surface and the base.

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Area of the Hemisphere Formula

To find the area of a hemisphere, we use the formula: \(2\pi r^2 + \pi r^2 = 3\pi r^2\), where \(r\) is the radius of the hemisphere. Now let's see how the formula is derived.

 

Derivation of the formula: The curved surface area of a hemisphere is half of the surface area of a sphere, which is \(2\pi r^2\). The base of the hemisphere is a circle, and its area is \(\pi r^2\). Adding the two areas gives the total surface area of the hemisphere: \(2\pi r^2 + \pi r^2 = 3\pi r^2\).

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How to Find the Area of a Hemisphere?

We can find the area of a hemisphere using the formula that involves the radius. The process is straightforward. Here is how to proceed: If the radius \(r\) is given, we find the area of the hemisphere using the formula \(3\pi r^2\).

 

For example, if the radius is 5 cm, what will be the area of the hemisphere? Area = \(3\pi r^2 = 3\pi \times 5^2 = 75\pi\). The area of the hemisphere is approximately 235.62 cm² using \(\pi \approx 3.14\).

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Unit of Area of Hemisphere

We measure the area of a hemisphere in square units. The measurement depends on the system used:

 

In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).

 

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

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Special Cases or Variations for the Area of Hemisphere

A hemisphere is a three-dimensional object with a unique shape. Here are some special cases to consider:

 

Case 1: Solid Hemisphere If only the curved surface area is needed, use the formula \(2\pi r^2\).

 

Case 2: Hollow Hemisphere If only the base area is considered, use the formula \(\pi r^2\).

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Tips and Tricks for Area of Hemisphere

To ensure accurate calculations for the area of a hemisphere, here are some tips and tricks you should know about:

 

  • Always ensure the radius is in the correct units for the area measurement.
     
  • Remember that a hemisphere includes both a curved surface and a flat base.
     
  • Use a calculator to evaluate \(\pi\) for more precise results.
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Common Mistakes and How to Avoid Them in Area of Hemisphere

Mistakes can occur when calculating the area of a hemisphere. Let's explore some common errors and how to prevent them.

Mistake 1

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Using the wrong formula

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Remember that the total area of a hemisphere includes both the curved surface and the base, so use \(3\pi r^2\).

Mistake 2

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Mixing up units

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Ensure that the radius is converted to the correct unit of measure before calculating the area.

Mistake 3

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Forgetting to include the base area

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A common mistake is to only calculate the curved surface area.

 

Remember to add the base area when finding the total area.

Mistake 4

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Incorrectly applying the formula for a full sphere

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Avoid using the formula for a full sphere (\(4\pi r^2\)) when calculating the area of a hemisphere.

 

Use \(3\pi r^2\) instead.

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Area of Hemisphere Examples

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Problem 1

A small bowl is in the shape of a hemisphere with a radius of 7 cm. What is the total surface area?

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We will find the area as 461.58 cm²

Explanation

Here, the radius \(r\) is 7 cm

The area of the hemisphere = \(3\pi r^2 = 3\pi \times 7^2 = 147\pi\).

The area is approximately 461.58 cm² using \(\pi \approx 3.14\).

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Problem 2

What is the curved surface area of a hemisphere with a radius of 10 m?

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We will find the curved surface area as 628 m²

Explanation

If only the curved surface area is needed, use the formula \(2\pi r^2\).

Here, the radius \(r\) is 10 m.

Hence, the curved surface area will be \(2\pi \times 10^2 = 200\pi\).

The curved surface area is approximately 628 m² using \(\pi \approx 3.14\).

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Problem 3

The base area of a hemisphere is 314 cm². What is the radius?

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We find the radius as 10 cm

Explanation

To find the radius, use the formula for the base area \(\pi r^2\).

Here, the base area is 314 cm². 314 = \(\pi r^2\). \(r^2 = \frac{314}{\pi}\). \(r^2 = 100\). \(r = 10\).

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Problem 4

Find the total surface area of a hemisphere with a radius of 3 cm.

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We will find the area as 84.78 cm²

Explanation

The given radius is 3 cm.

The total surface area of the hemisphere = \(3\pi r^2\).

Substituting the values: Area = \(3\pi \times 3^2 = 27\pi\).

The area is approximately 84.78 cm² using \(\pi \approx 3.14\).

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Problem 5

Help Sarah find the area of a hemisphere with a radius of 12 m.

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We will find the area as 1356.48 m²

Explanation

The radius is 12 m. Using the total surface area formula for the hemisphere:

Area = \(3\pi r^2 = 3\pi \times 12^2 = 432\pi\).

The area is approximately 1356.48 m² using \(\pi \approx 3.14\).

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FAQs on Area of Hemisphere

1.Is it possible for the area of a hemisphere to be negative?

No, the area of a hemisphere can never be negative. The area of any shape will always be positive.

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2.How to find the area of a hemisphere if only the radius is given?

If the radius is given, then use the formula \(3\pi r^2\) to find the total surface area.

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3.How to find the curved surface area of a hemisphere?

If the radius is given, use the formula \(2\pi r^2\) to find the curved surface area.

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4.How is the volume of the hemisphere calculated?

The volume of a hemisphere is calculated using the formula \( \frac{2}{3}\pi r^3\), where \(r\) is the radius.

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5.What is meant by the area of a hemisphere?

The area of a hemisphere is the total surface area, including both the curved surface and the flat base.

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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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Fun Fact

: She has songs for each table which helps her to remember the tables

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