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Last updated on September 13, 2025
Area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of the semicircle.
A semicircle is a two-dimensional shape that forms half of a circle. The area of the semicircle is the total space it encloses. Understanding the properties of a circle helps in calculating the area of a semicircle.
To find the area of the semicircle, we use the formula: \( \frac{\pi r^2}{2} \), where \( r \) is the radius of the circle from which the semicircle is formed. Now let’s see how the formula is derived.
Derivation of the formula:- The area of a full circle is \(\pi r^2\). Since a semicircle is half of a circle, its area is half of the full circle's area. Therefore, the area of the semicircle = \(\frac{\pi r^2}{2}\).
We can find the area of the semicircle using the radius of the circle. The formula is straightforward since a semicircle is simply half of a circle. Let’s discuss how to use the formula:
If the radius \( r \) is given, we find the area of the semicircle using the formula: Area = \(\frac{\pi r^2}{2}\). For example, if the radius is 10 cm, what will be the area of the semicircle? Area = \(\frac{\pi ×10^2}{2}\) = \(\frac{\pi × 100}{2}\) = \(50\pi \approx 157.08\) The area of the semicircle is approximately 157.08 cm\(^2\).
We measure the area of a semicircle in square units. The measurement depends on the system used: In the metric system, the area is measured in square meters (m\(^2\)), square centimeters (cm\(^2\)), and square millimeters (mm\(^2\)).
In the imperial system, the area is measured in square inches (in\(^2\)), square feet (ft\(^2\)), and square yards (yd\(^2\)).
The area of a semicircle is straightforward to calculate once the radius of the circle is known. Take a look at the special cases:
Case 1: Using the radius If the radius is given, use the formula Area = \(\frac{\pi r^2}{2}\), where \( r \) is the radius.
Case 2: Using the diameter If the diameter is given, first find the radius by dividing the diameter by 2, and then use the formula Area = \(\frac{\pi r^2}{2}\).
To ensure correct results while calculating the area of the semicircle, here are some tips and tricks you should know about:
It is common for students to make mistakes while finding the area of the semicircle. Let’s take a look at some mistakes made by students.
A semicircular window has a radius of 7 m. What will be the area?
We will find the area as approximately 76.97 m\(^2\).
Here, the radius \( r \) is 7 m.
The area of the semicircle = \(\frac{\pi × 7^2}{2}\) = \(\frac{\pi × 49}{2}\) = \(24.5\pi \approx 76.97\) m\(^2\).
What will be the area of the semicircle if the diameter is 16 cm?
We will find the area as approximately 100.53 cm\(^2\).
If the diameter is given, first find the radius by dividing the diameter by 2.
Here, the diameter is 16 cm, so the radius is 8 cm.
The area of the semicircle = \(\frac{\pi × 8^2}{2}\) = \(\frac{\pi × 64}{2}\) = \(32\pi \approx 100.53\) cm\(^2\).
The area of a semicircular garden path is 157 m\(^2\). What is the radius?
We find the radius as approximately 10 m.
To find the radius, use the formula \( \frac{\pi r^2}{2} = 157 \).
Rearrange to find \( r \): \(\pi r^2 = 314\) \(r^2 = \frac{314}{\pi}\) \(r \approx \sqrt{\frac{314}{3.14159}} \approx 10\) m.
Find the area of the semicircle if its radius is 12 cm.
We will find the area as approximately 226.2 cm\(^2\).
The given radius is 12 cm.
The area of the semicircle = \(\frac{\pi × 12^2}{2}\) = \(\frac{\pi × 144}{2}\) = \(72\pi \approx 226.2\) cm\(^2\).
Help Lisa find the area of the semicircle if the diameter is 20 m.
We will find the area as approximately 157 m\(^2\).
The diameter is 20 m, so the radius is 10 m.
The area of the semicircle = \(\frac{\pi × 10^2}{2}\) = \(\frac{\pi × 100}{2}\) = \(50\pi \approx 157\) m\(^2\).
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.
: She has songs for each table which helps her to remember the tables