# Segment of a Circle

Segment of a Circle
Go back to  'Circles'

In this lesson, we will expand our knowledge about cirles by learning about the segment of a circle, which will include its mathematical definition, area of segment, types of segments in a circle, theorems based on segment of a circle and related real life word problems and solutions.

A segment of a circle is the region enclosed by a chord and an arc so formed touching the end points of the chord.

In the simulation given below, drag the end point of the chord to form segment of a circle.

Isn't it interesting? Let's explore further.

## What is Segment of a Circle?

Cersie and her brother had a chocolate pie that they had to distribute equally among themselves. But, Cersei was clever to cut it in such a way that she got the larger slice.

She did not cut it through the centre of the circular pie, but rather she cut it through another line known as chord of a circle to create segments of pie.

Now let us look at the definition of segment of a circle.

### Definition

A segment of a circle is the region enclosed by a chord and an arc so formed touching the end points of the chord.

If we break a circle into two parts by a straight line touching the circumference of the circle, those two parts are called segments of the circle.

## What is the Area of a Segment?

Area of a segment is the area enclosed between the chord and the minor or major arc of the circle.

We find the area of a segment with the help of central angle formed by the chord and radius of the circle (denoted as $$\theta$$ in the figure given below)

Area of the Segment can be found by taking the differance of the Area of the sector and the Area of the triangle enclosed in it. Or else, it can be found by subtracting the area of the other given segment (major or minor) from the area of the circle.

## Types of Segment in a Circle?

According to the area enclosed by the segments, it can be classified into two types: major segment and minor segment. The major segment covers a larger portion of the circle, while the minor segment covers a smaller portion of the circle.

## Area Formula of Segment of a Circle

 $$A= \frac{1}{2} \times \ r^2 \ (\theta-\sin\ \theta)$$
 Formula to be used to find Area of Segment Area of a Segment in Radians $$A= \frac{1}{2} \times \ r^2 \ (\theta-\sin\ \theta)$$ Area of a Segment in Degrees $$A= \frac{1}{2} \times \ r^2 \ [\frac{\pi}{180^o}(\theta-\sin\ \theta)]$$

### Segment of a Circle Calculator

In the simulation below, drag points P and Q to change the radius and central angle and carefully observe the area of segment so formed.

## Theorems on Segment of a Circle

Mainly, there are two theorems based on the segment of a Circle.

1. Angles in the same segment theorem
2. Alternate segment theorem

### Angles in the Same Segment Theorem

It states that angles formed in the same segment of a circle are always equal.

### Alternate Segment Theorem

This theorem states that the angle formed by the tangent and the chord at the point of contact is equal to the angle formed in the alternate segment on the circumference of the circle through the end points of the chord.

Important Notes
1. The segment formed by the diameter of a circle with its corresponding arc is the largest possible segment in any circle.
2. Based on the area, there are two types of Segment- Minor Segment and Major segment.
3. Area of minor segment = Area of circle - Area of major segment
4. Area of Segment = Area of the corresponding sector - Area of the triangle (formed by the radii and the chord in the same sector)

## Solved Examples

 Example 1

In a pizza slice, if the central angle is 60 degrees and the length of its radius is 4 units, then find the area of the segment formed if we remove the triangle part out of the pizza slice.

Solution

Formula of Area of Segment of a circle, $$A= \frac{1}{2} \times \ r^2 \ [\frac{\pi}{180^o}(\theta-\sin\ \theta)]$$

$$A= \frac{1}{2} \times \ 4^2 \ [\frac{\pi}{180^o}(\ 60-\sin\ \ 60)]$$

$$A= 8 \ [\frac{\pi}{3}-\ \frac{\pi}{180^o} \times \frac{\sqrt{3}}{2}]$$

$$A= \frac{\ 8\pi}{3}-\ \frac{\pi\sqrt{3}}{45}\ sq\ units$$

 $$\therefore The\ Area\ of\ the\ Segment\ is\ \frac{\ 8\pi}{3}-\ \frac{\pi\sqrt{3}}{45}\ sq\ units$$
 Example 2

Find the area of the segment formed by the minute and hour hand of a clock at 3 PM, if the length of both the hands is 4 inches.

Solution

Angle formed at 3 PM in a clock is $$90^o$$.

Area of the Segment, $$A= \frac{1}{2} \times \ r^2 \ [\frac{\pi}{180^o}(\theta-\sin\ \theta)]$$

$$A= \frac{1}{2} \times \ 4^2 \ [\frac{\pi}{180^o}(\ 90^o-\sin\ \ 90^o)]$$

$$A= 8 \ [\frac{\pi}{2}-\frac{\pi}{180}]$$

$$A= \frac{178 \pi}{45} \ sq.\ inches$$

 $$\therefore\ The\ Area\ of\ Segment\ is\ \frac{178 \pi}{45} \ sq.\ inches$$
 Example 3

Find the area of the red coloured portion if the radius of the wheel is 8 inches and the measurement of central angle is 45 degrees.

Solution

Area of Segment of Wheel, $$A= \frac{1}{2} \times \ r^2 \ (\theta-\sin\ \theta)$$

$$A= \frac{1}{2} \times \ 8^2 \ (\frac{\pi}{4}-\frac{1}{\sqrt{2}})$$

$$A= 32 \ (\frac{\pi}{4}-\frac{1}{\sqrt{2}})$$

$$A= 8\pi-16{\sqrt{2}} \ sq.\ inches$$

 $$\therefore The\ Area\ of\ Segment\ is\ 8\pi-16{\sqrt{2}} \ sq.\ inches$$
 Example 4

If the area of a sector is 100 sq. ft and the area of the enclosed triangle is 78 sq. ft, what is the area of the segment?

Solution

Area of the Segment= Area of the Sector- Area of the Triangle

= 100 sq. ft.- 78 sq. ft.

= 22 sq. ft.

 $$\therefore The\ Area\ of\ Segment\ is\ 22 \ sq.\ ft.$$
 Example 5

Find the area of the major segment of a circle if the area of corresponding minor segment is 62 sq. units and the radius is 14 units.

Solution

Area of the major segment= Area of the circle- Area of the minor Segment

Area of major Segment= $$\pi r^2- 62$$

= $$\frac{22}{7} \times 14 \times 14 - 62$$

= $$554\ sq.\ units$$

 $$\therefore The\ Area\ of\ major\ Segment\ is\ 554 \ sq.\ units$$

Challenging Question
1. Can you find the area of all the segments formed by a regular hexagon inside a circle of radius 5 units.

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of segment of a circle. The math journey around segment of a circle starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

## About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is the Area of a Sector?

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

## 2. What is the difference between a sector of a circle and a segment of a circle?

A sector of a circle is the region enclosed by two radii and the corresponding arc, while a segment of a circle is the region enclosed by a chord and the corresponding arc.

## 3. Is a semicircle a segment?

A semicircle is the largest segment in any circle formed by the diameter and the corresponding arc.

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