CBSE Class 10 Maths Notes: Areas Related to Circles

**Areas Related to Circles: A Comprehensive Guide**

Welcome to the exploration of ‘Areas Related to Circles’! This chapter delves into the fascinating world of circles, their sectors, and segments, equipping you with the knowledge to calculate areas and perimeters in various scenarios. Let’s embark on this mathematical journey!

**Area of a Sector: Derivation and Formula**

A sector of a circle is a region enclosed by an arc and two radii. Understanding the area of a sector is fundamental. Let’s break it down:

Definitions

• **Sector:** The region enclosed by an arc of a circle and the two radii joining the center to the endpoints of the arc.
• **Central Angle (θ):** The angle formed at the center of the circle by the two radii.

Core Principles

The area of a sector is directly proportional to the central angle. If the central angle is a full circle (360°), the sector’s area is the entire circle’s area.

Formulae

The area of a sector can be calculated using the following formula:

Area of Sector = $\frac{\theta}{360} \times \pi r^2$

where:

• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

or

Area of Sector = $\frac{1}{2}lr$

where:

• $l$ is the length of arc
• $r$ is the radius of the circle

**Area of a Segment: Derivation and Formula**

A segment of a circle is the region enclosed by an arc and a chord. Finding the area of a segment involves a slightly different approach.

Definitions

• **Segment:** The region enclosed by a chord and the corresponding arc.
• **Chord:** A line segment joining two points on the circle.

Core Principles

The area of a segment is found by subtracting the area of the triangle formed by the chord and the two radii from the area of the corresponding sector.

Formulae

The area of a segment is given by:

Area of Segment = Area of Sector – Area of Triangle

Area of Triangle will depend on the angle at the center of the circle. We will cover this for the central angles of 60°, 90°, and 120°.

Specifically:

• **For 60°:** Area of Segment = $\frac{\theta}{360} \times \pi r^2 – \frac{\sqrt{3}}{4}r^2$
• **For 90°:** Area of Segment = $\frac{\theta}{360} \times \pi r^2 – \frac{1}{2}r^2$
• **For 120°:** Area of Segment = $\frac{\theta}{360} \times \pi r^2 – \frac{1}{2}r^2\sin(120)$

**Problems on Area and Perimeter/Circumference**

Let’s put our knowledge to the test with some example problems.

Example 1: Area of a Sector

Find the area of a sector of a circle with a radius of 6 cm if the angle of the sector is 60°.

Solution: Area = $\frac{60}{360} \times \pi \times 6^2 = 6\pi cm^2$

Example 2: Area of a Segment (60°)

Find the area of the segment formed by a chord which forms a $60^{\circ}$ angle at the center of a circle of radius $12$ cm

Solution: Area of Segment = $\frac{60}{360} \times \pi \times 12^2 – \frac{\sqrt{3}}{4} \times 12^2 = 24\pi – 36\sqrt{3} cm^2$

Example 3: Area of a Segment (90°)

Find the area of the segment formed by a chord which forms a $90^{\circ}$ angle at the center of a circle of radius $7$ cm

Solution: Area of Segment = $\frac{90}{360} \times \pi \times 7^2 – \frac{1}{2} \times 7^2 = \frac{49\pi}{4} – \frac{49}{2} cm^2$

Example 4: Perimeter of Sector

Find the perimeter of a sector of a circle with radius 10 cm and a central angle of $60^\circ$

Solution: Arc length = $\frac{60}{360} \times 2\pi \times 10 = \frac{10\pi}{3}$ cm. Perimeter = Arc length + 2 * radius = $\frac{10\pi}{3} + 20$ cm

**Tips and Tricks**

• Remember the relationship between angles and areas – larger angles mean larger sectors.
• Always draw a diagram to visualize the problem.
• Pay close attention to units!