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Last updated on September 15, 2025
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.
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.
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: \(2\pi r^2 + \pi r^2 = 3\pi r^2\).
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 \(\pi \approx 3.14\).
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).
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 \(\pi r^2\).
To ensure accurate calculations for the area of a hemisphere, here are some tips and tricks you should know about:
Mistakes can occur when calculating the area of a hemisphere. Let's explore some common errors and how to prevent them.
A small bowl is in the shape of a hemisphere with a radius of 7 cm. What is the total surface area?
We will find the area as 461.58 cm²
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 \(\pi \approx 3.14\).
What is the curved surface area of a hemisphere with a radius of 10 m?
We will find the curved surface area as 628 m²
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 \(\pi \approx 3.14\).
The base area of a hemisphere is 314 cm². What is the radius?
We find the radius as 10 cm
To find the radius, use the formula for the base area \(\pi r^2\).
Here, the base area is 314 cm². 314 = \(\pi r^2\). \(r^2 = \frac{314}{\pi}\). \(r^2 = 100\). \(r = 10\).
Find the total surface area of a hemisphere with a radius of 3 cm.
We will find the area as 84.78 cm²
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 \(\pi \approx 3.14\).
Help Sarah find the area of a hemisphere with a radius of 12 m.
We will find the area as 1356.48 m²
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 \(\pi \approx 3.14\).
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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