Areas Related to Circles: Formulas, Concepts & Examples

Areas related to circles is an important topic in geometry that helps students understand how to calculate the area of the space occupied by different parts of a circle, such as sectors, segments, and semicircles. It is built on the basic concept of the area of a circle and extends it to other real-world shapes formed within or from a circle. This topic is widely used in solving practical problems involving arcs, chords, and circular regions.

Table of Contents

• Area of a Circle
• Area of a Semicircle
• Area of a Sector of a Circle
• Area of a Segment of a Circle
• Frequently Asked Questions on Areas Related to Circles

Area of a Circle

A circle is a simple closed two-dimensional figure in which all points on its boundary are at a fixed distance from a fixed point called the centre. The area of a circle is the region enclosed within its boundary.

Area of a circle =  $πr^{2}$ Where, r = radius of circle

Area of a Semicircle

A semicircle is formed when a circle is cut in half along its diameter. The area of a semicircle is half the area of the circle.

Area of semicircle =  $\frac{1}{2}πr^{2}$ Where, r = radius of circle

Area of a Sector of a Circle

A sector of a circle is a region of a circle enclosed by two radii and the arc between them. The two radii form the angle at the centre of the circle known as the central angle. This central angle determines the size of the sector.

Let r be the radius of a circle. We know that the area of a circle is  $πr^{2}$, i.e., when the degree measure of the angle at the centre is 360, the area of the sector =  $πr^{2}$.

Then, when the degree measure of the angle at the centre is 1, the area of the sector =  $\frac{πr^{2}}{360}$

∴ when the degree measure of the angle at the centre is θ, =  $\frac{πr^{2}}{360}×θ$ =  $\frac{θ}{360}× πr^{2}$

Area of sector =  $\frac{θ}{360}×πr^{2}$

Where, r = radius of circle

θ = angle of the sector in degrees

NOTE: Length of an arc of a sector of a circle with radius r and angle with degree measure is \frac{θ}{360}×2πr

Area of a Segment of a Circle

A segment of a circle is the region formed when a circle is divided by a chord. It is the part of the circle enclosed between a chord and the corresponding arc.

APB = Area of the sector OAPB – Area of ∆ OAB

Area of ∆OAB =  $\frac{1}{2}r^{2}sinθ$

=  $r^{2}\left[\frac{πθ}{360} - \frac{sinθ}{2}\right]$

Area of the segment =  $r^{2}\left[\frac{πθ}{360} - \frac{sinθ}{2}\right]$

Where, r = radius of circle

θ = angle of the sector in degrees

Example 1: Find the area of a circle with radius 7 cm. (Use π = 22/7) Solution: Given, r = 7 cm Area of a circle =  $πr^{2}$

Area=(22/7)​×7×7=154  $cm^{2}$

Example 2: Find the area of a semicircle with radius 14 cm. (Use π = 22/7) Solution: Given, r = 14 cm Area of a semicircle =  $\frac{1}{2}πr^{2}$

Example 3: Find the area of a sector with radius 7 cm and angle 60°.

Solution: Given, r = 7 cm and θ = 60°

Area of sector = 25.67  $cm^{2}$

Solution: Given r = 14 cm and θ = 60°

Area of the segment =  $r^{2}\left[\frac{πθ}{360} - \frac{sinθ}{2}\right] = (14)^{2}\left[\frac{π × 60}{360} - \frac{sin60}{2}\right]$

=  $(14)^{2}\left[\frac{π × 60}{360} - \frac{\frac{\sqrt{3}}{2}}{2}\right]$

= 17.8  $cm^{2}$