Area of a Semicircle
The area of a semicircle is defined as the amount of region present inside it. Alternatively, the area of a semicircle represents the total number of unit squares that can be fit in it. You must have observed semicircle shape in our day-to-day lives, for example, the shape of a protractor, railway tunnel, etc resemble a semicircle in a 2-D plane. Let us understand the formula and method to calculate the area of a semicircle using solved examples in the following sections.
|1.||What Is Area of a Semicircle?|
|2.||Formula of Area of Semicircle|
|3.||Derivation of Area of a Semicircle Formula|
|4.||How to Calculate the Area of a Semicircle?|
|5.||FAQs on Area of Semicircle|
What Is 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 using measurement units like in2, cm2, m2, yd2, ft2, etc.
General Formula of Area of Semicircle
The region 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. A semicircle's area can be calculated using radius as well as diameter. The formula to calculate the area of a semicircle in terms of radius 'r' is given as,
Area of Semicircle = πr2/2
The formula 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
Let's understand how the formula for the area of a semicircle is derived. Here's a figure for your reference to know more about the concept of this formula.
- In the above figure, 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 Calculate the Area of a Semicircle?
The steps that can be followed to calculate the area of a semicircle using the length of either the diameter of the radius are given as,
- Step 1: Note down the measure of the given radius or diameter of the semicircle.
- Step 2: Apply any of the general formulas used to find a semicircle's area.
Area of semicircle = πr2/2 = πd2/8, where 'r' is the radius, and 'd' is the diameter.
- Step 3: Mention the area in square units.
Example: If the radius of a semicircle is 14 inches, calculate its area.
Area of Semicircle = πr2/2 = [(22/7) × 14 × 14]/2 = 308 in2
Now, let's find the area of a semicircle with a diameter of 7 inches.
Area of Semicircle = πd2/8 = [(22/7) × 7 × 7]/8 = 77/4 in2 = 19.25 in2
- The semicircle is one-half of a circle.
- The area of a circle, with radius 'r', is πr2.
- The area of a semicircle, with radius 'r', is πr2/2.
Solved Examples on Area of Semicircle
Example 1: Calculate the area of the semicircle whose diameter is 12 inches. Express your answer in terms of π.
The radius of a semicircle is half of its diameter.
Radius, r = d/2 = 12/2 = 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 = 36π2/2 = 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)
If a circle is divided into two equal parts, the two parts are semicircle. The radius of the 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) = 3928.57yd2
Answer: Area covered by Sam is 3928.57yd2
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)
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, use the formula of area of the semicircle and equate 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 = √31.818
⇒ r = 5.64 in
The diameter of the semicircle is twice its radius. Therefore, the diameter of each slice, d = 11.28 in.
Answer: The diameter of each slice is 11.28 in.
FAQs on Area of Semicircle
What Is Meant by Area of Semicircle?
The area of the semicircle is defined as the total region encompassed by the semi-circle in the 2-D plane. Alternatively, it is the region covered inside the perimeter of a semicircle.
How Do You Find the Perimeter and Area of a Semicircle?
The area of a semicircle can be calculated by dividing the area of a circle by 2. While its perimeter is the sum of half of the circumference of a circle and the diameter.
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. The formula to calculate the area of the semicircle is given as, Area = πr2/2 = πd2/8, where 'r' is the radius, and 'd' is the diameter.
What Unit Is Used to Express the Area of Semicircle?
We use square units or unit2 to represent the area of a semicircle. The common units that can be used to calculate the area of a semicircle in2, m2, cm2, yd2, ft2, etc.
How to Find the Area of Semicircle Using Diameter?
The area of a semicircle can be calculated given the diameter of the semicircle 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?
The formulas to calculate the area of semicircle using radius and diameter are given as, 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.
How Does Area of Semicircle Change When the Radius is Doubled?
The area of a semicircle is proportional to the square of its radius. Therefore, the area quadruples when the radius of the semicircle is doubled.