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Area of a Semicircle

The area of a semicircle is defined as the space occupied by the semi circle. Alternatively, the area of a semicircle represents the total number of unit squares that can fit in it. A semicircle shape can be commonly seen in our day-to-day lives, for example, the shape of a protractor, or a railway tunnel, etc., resembles a semicircle in a 2-D plane. Since semi circle represents half a circle, the area of a semi circle is the area of half circle. Let us learn how to find the area of a semi-circle, the area of semicircle formula and the method to find the area of half circle using solved examples in the following sections.

1.	What is the Area of a Semicircle?
2.	Area of Semicircle Formula
3.	Derivation of Area of a Semicircle Formula
4.	How to Find the Area of a Semicircle?
5.	FAQs on Area of Semicircle

What is the Area of a Semicircle?

The area of a semicircle is the amount of space enclosed within the boundary of a semicircle. Observe the following figure which shows that the colored region within the boundary of the semi-circle is the area occupied by it. The area of a semicircle is expressed in square units like in², cm², m², yd², ft², etc.

Area of semicircle formula

Area of Semicircle Formula

The region enclosed within the boundary of the semicircle is the area occupied by the semicircle. A semicircle is half of a circle. So, the area of a semicircle will be half the area of a circle. The area of a semicircle can be calculated using the radius as well as the diameter. The formula which is used to calculate the area of a semicircle in terms of radius 'r' is expressed as,
Area of Semicircle = πr²/2

The formula which is used to calculate the area of a semicircle in terms of diameter 'd' is,
Area of Semicircle = πd²/8

π is a constant whose value is 22/7 or 3.14

Derivation of Area of a Semicircle Formula

The derivation of the area of a semicircle can be understood with the help of the following figure.

Area of Semicircle Derivation

In the figure given above, observe how the circle converts into a triangle, and how the radius becomes the height of the triangle, while the circumference, which is 2πr, becomes its base.
We know that the area of a triangle is found by multiplying its base by the height and then dividing by 2, which is
(1/2) × 2πr × r
After simplifying this, we get the area of the circle as πr²
Area of Circle = πr²
Now the area of the semicircle is half the area of the circle.
Therefore, the area of the semicircle is πr²/2
Area of Semicircle = πr²/2

How to Find the Area of a Semicircle?

The following steps can be followed to find the area of a semicircle using the length of either the diameter or the radius, whichever is given:

Step 1: Note down the measure of the radius or diameter of the semicircle, whichever is given. Use the value of π = 22/7 or 3.14
Step 2: Apply the appropriate formula to find the area of a semicircle. For example, if the radius is given, then we use the formula, Area of semicircle = πr²/2, where 'r' is the radius, and when the diameter is given then we use the formula, Area of semicircle = πd²/8, where 'd' is the diameter.
Step 3: Solve the equation and mention the area in square units.

Example: If the radius of a semicircle is 14 inches, calculate its area.

Solution: Area of Semicircle = πr²/2 = [(22/7) × 14 × 14]/2 = 308 in²

Now, let us find the area of a semicircle when the diameter is given.

Example: If the diameter of a semicircle is 28 inches, find its area.

Solution: Area of Semicircle = πd²/8 = [(22/7) × 28 × 28]/8 = 77/4 in² = 308 in²

Important Notes on Area of Half Circle

The semicircle is one-half of a circle. So, the area of a semicircle is actually the area of half circle.
The area of a circle, in terms of radius 'r' is πr².
The area of a semicircle, with radius 'r' is πr²/2.

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Examples on Area of Semicircle

Example 1: Calculate the area of a semicircle whose radius is 6 inches. Express your answer in terms of π.

Solution:

Radius, r = 6 in

Now, let us calculate the area of the semicircle by using the formula, A = πr²/2

⇒ Area = πr²/2 = π(6)²/2 = 36π/2 = 18π in²

Answer: Area of the semicircle is 18π in²
Example 2: Sam and his friend James decided to grow grass in a circular park of a radius of 50 yards. If both of them covered equal areas, how much area did Sam cover to grow the grass? (Use π = 22/7)

Solution:

If a circle is divided into two equal parts, the two parts are semicircle. The radius of the two semicircles is equal to the radius of the circle. Thus, the area covered by Sam is the area of a semicircle.

Area = πr²/2 = π(50)²/2 = (2500/2) × (22/7) = 3928.57 yd²

Answer: Area covered by Sam is 3928.57 yd²
Example 3: Maria and her friend Nia ordered a pizza on a Friday night. The area of the pizza was 100 in². Nia cut the pizza into two equal slices along the diameter. Calculate the diameter of each semi-circular slice. (Use π = 22/7)

Solution:

The area of each slice is equal to half the area of the whole pizza. Thus, the area of each slice = 100/2 in² = 50 in²

Now, we will use the formula of area of the semicircle and equate it with the slice area, that is 50 in², and solve for the radius 'r'.

⇒ πr²/2 = 50
⇒ (22/7) r² = 100
⇒ r² = 700/22
⇒ r = √31.81
⇒ r = 5.64 in

The diameter of the semicircle is twice its radius. Therefore, the diameter of each slice, d = 2 × radius = 2 × 5.64 ⇒ 11.28 in.

Answer: The diameter of each slice is 11.28 in.

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Practice Questions on Area of Semicircle

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FAQs on Area of a Semicircle

What is the Area of Semicircle?

The area of a semicircle is defined as the total region enclosed by the semi-circle in the 2-D plane. Since a semicircle is half of a circle, the area of a semicircle is half the area of a circle.

How to Find the Perimeter and Area of a Semicircle?

The perimeter of a semicircle is half of the circumference of a circle plus the length of the diameter. It can be expressed as, Perimeter of semicircle (in terms of diameter) = πr + d and Perimeter of semicircle (in terms of radius) = r(π + 2). The area of a semicircle can be calculated by dividing the area of a circle by 2. This can be expressed as, Area of a semicircle (in terms of radius) = πr²/2

What is the Area of Semicircle Formula?

The area of a semicircle can be calculated using the length of radius or diameter of the semicircle, whichever is given. The formula which is used to calculate the area of the semicircle is expressed as, Area of semicircle (when the radius is given) = πr²/2, Area of semicircle (when the diameter is given) = πd²/8, where 'r' is the radius, and 'd' is the diameter.

What Unit is Used to Express the Area of Half Circle?

The area of a semicircle is expressed in square units. The common units that are used to express the area of a semicircle are in², m², cm², yd², ft², etc.

How to Find the Area of Semicircle Using Diameter?

If the diameter of the semicircle is given, then the area of the semicircle can be calculated using the formula, Area of semicircle using diameter = πd²/8, where 'd' is the diameter.

How to Find the Radius and Diameter From the Area of a Semicircle?

When the area of a semicircle is given, the radius and the diameter can be calculated using the same formulas: Area of a semicircle with radius = πr²/2; Area with diameter = πd²/8, where 'r' is the radius, and 'd' is the diameter. We can substitute the known value of the area in these formulas and find the missing radius and diameter.

What is the Area of a Semicircle with a Diameter of 8 cm?

The area of a semicircle can be calculated using the formula, Area of semicircle = πd²/8, where 'd' is the diameter. After substituting the value of d = 8, we get, Area of semicircle = (π × 8²)/8 = (3.14 × 8²)/8 = 25.12 cm²

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