What is the definition of a semicircle?
When a line passing through the center touches the circle, this line segment which is formed is called the diameter. The diameter cuts the circle into identical halves. This plane figure thus formed is called a semicircle.
The area of a semicircle
The region or inner space of a circle is referred to as its area. A semicircle, as we know, is half of a circle, and therefore its area will also be half that of a circle’s area.
The area of a circle is \(\pi r^2\), where r is the radius of the circle. Therefore, the area of a semicircle is \(=\frac{\pi r^2}{2}\).
- r is the radius of a semicircle.
- \(\pi\)(pi) is roughly \(\frac{22}{7}\)or 3.142.
The circumference of a semicircle
The circumference of a semicircle is equal to half of the circumference of a circle. The measurement of the arc that forms a semicircle is the circumference of a semicircle. The formula for the circumference of a circle is 2\(\pi r\), where r is the radius.
As a result, the circumference of a semicircle is equal to
\(=\frac{2\pi r}{2}\)
\(=\pi r\) units.
The Perimeter of a semicircular region
A semicircular closed region has a perimeter equal to half of the circumference of a circle plus its diameter. The circumference of a circle is 2\(\pi r\) or \(\pi\)d.
A perimeter of a semicircle is = \(\left(\frac{1}{2}\right)\pi\)d + d
= \(\pi\)r + 2r
Where,
- r is the radius of the semicircle.
- d is the diameter of the semicircle.
- \(\pi\)(pi) is roughly \(\frac{22}{7}\), or 3.142.
List of the formulas of a semicircle
Area | \(\frac{\pi r^2}{2}\) |
|---|---|
Perimeter | \(\left(\frac{1}{2}\right)\pi d+d\) when the diameter (d) is known. \(\pi\)r + 2r; when the radius (r) is known. |
Circumference | \(\pi\)r |
Angle formed in a semicircular segment. | 90 degrees, that is, a right angle . |
Central angle | 180 degrees |
Solved Examples
Example 1:
Calculate the area of a semicircle with a radius of 14 cm.
Solution:
The radius of the semicircle (r) = 14 cm.
Now, area of the semicircle = \(\frac{\pi r^2}{2}\)
= \(\frac{1}{2}\times\frac{22}{7}\times14\times14\)
= 308
The semicircle has an area of 308 centimeter square.
Example 2:
Find the perimeter of a semicircle that has a diameter of 77cm.
Solution:
The diameter of the semicircle is 77 centimeters.
\(=\left(\frac{1}{2}\right)\pi d+d\) Formula for the perimeter of the semicircle.
\(=\frac{1}{2}.\left(\frac{22}{7}\right).77+77\) Substitute \(\frac{22}{7}\) for \(\pi\) and 77 for d.
= 198 Simplify.
The perimeter of the semicircle is 198 centimeters.
Example 3:
The radius of the semicircle is 14 meters. Find the circumference of the semicircle.
Answer:
The radius of the semicircle is 14 meters.
The circumference of a semicircle is = \(\pi\)r
\(=\left(\frac{22}{7}\right).14\) Substitute \(\frac{22}{7}\) for \(\pi\) and 140 for r.
= 44 Multiply.
The circumference of the semicircle is 44 meters.
When a circle is divided into equal parts by its diameter, two semicircles are formed. As a result, every circle has two semicircles, the sum of whose areas equals the area of the circle.
A quarter-circle is half of a semicircle. The closed figure formed with the quarter circle and the two radii is called a quadrant. The formulas for the quadrant are as follows:
Circumference of a quarter circle = \(\frac{\pi r}{2}\)
The perimeter of a quadrant = \(\frac{\pi r}{2}+2r\)
The area of a quadrant = \(\frac{\pi r^2}{4}\)
The diameter of a semicircle is twice its radius, just like a circle. If we cut a circle into two equal parts to make a semicircle, the diameter of the circle and the two semicircles formed are the same.
The area of a semicircle is the area within the boundary of a semicircle. It is the exact half of the are of a circle, which is \(\pi r^2\), where r is the radius of the circle.
The perimeter of a semicircle is not half that of a full circle. The perimeter of a semicircle is equal to the sum of the arc length and the diameter. As a result, the formula for calculating the perimeter of a semicircle is πr + 2r units, where r is the radius of the semicircle.