What is a Segment of a Circle?(Definition, Formulas & Examples)

The term segment is used for each of the parts into which something is or may be divided. In circles also the parts of a circle when divided by a chord is called a segment of a circle. Let’s understand what is a segment of a circle and how to calculate its length and area.

What is a Segment of a Circle?

A segment of a circle is a region bounded by a chord and a corresponding arc lying between the chord’s endpoints. We can also say that segment of a circle is a region of a circle that is created by breaking apart from the rest of the circle through a secant or a chord. 

Types of Segments of a Circle

Whenever a chord is drawn in a circle, two types of segments are formed. 

These two segments are called

Minor Segment: The segment having a smaller area is known as the minor segment. In the above figure, $\text{APB}$ is a minor segment.

Major Segment: The segment having a larger area is known as the major segment. In the above figure, $\text{AQB}$ is a major segment.

Note: If major or minor is not specified with a segment, it means minor segment.

Properties of Segment of Circle

The following are the properties of a segment of a circle.

• It is the area that is enclosed by a chord and an arc.
• The angle subtended by the segment at the center of the circle is the same as the angle subtended by the corresponding arc. This angle is usually known as the central angle.
• A minor segment is obtained by removing the corresponding major segment from the total area of the circle.
• A major segment is obtained by removing the corresponding minor segment from the total area of the circle.
• A semicircle is the largest segment in any circle formed by the diameter and the corresponding arc.

Area of Segment of a Circle

The area of a segment of a circle is given by the formula $\text{A} = \frac{1}{2} \times (\theta – \sin \theta) \times r^2$.

Consider a circle with centre $\text{O}$ and radius $r$. Further consider two points $\text{A}$ and $\text{B}$ on the circumference of a circle, such that $\text{APB}$ is a chord and $\angle \text{AOB} = \theta$ is a central angle.

The two radii($\text{OA}$ and $\text{OB}$) and the chord($\text{AB}$) of the segment together form a triangle($\triangle \text{AOB}$). 

Thus, the area of a segment of a circle is obtained by subtracting the area of the triangle from the area of the sector. i.e., $\text{Area of a segment of circle } = \text{ Area of the sector } – \text{ Area of the triangle}$

Area of sector $\text{OAPB}$ = $\frac{\theta \pi r^2}{360}$

To find the area of $\triangle \text{OAB}$, let’s consider it separately.

In the above figure $\triangle \text{AOB}$ is an isosceles triangle, with sides $\text{OA} = \text{OB} = r$ (Radius of a circle). Let’s draw a perpendicular from vertex $\text{O}$ on the opposite side $\text{AB}$ at a point $\text{M}$.

Then $\angle \text{AOM} = \angle \text{MOB} = \frac{\theta}{2}$ and $\text{AM} = \text{MB}$. (Perpendicular drawn from a vertex of an isosceles triangle bisects the angle and opposite side)

Area of $\triangle \text{AOB} = \frac{1}{2} \times \text{AB} \times \text{OM}$ $\left(\frac{1}{2} \times \text{ Base } \times \text{ Altitude} \right)$

Now in right triangle $\text{OMB}$, 

$\sin \frac{\theta}{2} = \frac{\text{MB}}{\text{OB}}$

$=> \sin \frac{\theta}{2} = \frac{\text{MB}}{r}$

$=> \text{MB} = r \sin \frac{\theta}{2}$

And, $\cos \frac{\theta}{2} = \frac{\text{OM}}{\text{OB}}$

$=> \cos \frac{\theta}{2} = \frac{\text{OM}}{r}$

$=> \text{OM} = r \cos \frac{\theta}{2}$

Also, $\text{AB} = 2 \text{MB}$

Therefore $\text{AB} = 2 r \sin \frac{\theta}{2}$

Thus area of $\triangle \text{AOB} = \frac{1}{2} \times 2 r \sin \frac{\theta}{2} \times r \cos \frac{\theta}{2}$

$= r \sin \frac{\theta}{2} \times r \cos \frac{\theta}{2}$

$= r^2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}$

Hence area of segment $\text{AOBP} = \frac{\theta \pi r^2}{360} – r^2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}$

$= r^2 \left(\frac{\theta \pi}{360} – \sin \frac{\theta}{2} \cos \frac{\theta}{2} \right)$

$= r^2 \left(\frac{\theta \pi}{360} – \frac{\sin \theta}{2} \right)$ $\left(\sin \theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \right)$

Examples on Area of Segment of a Circle

Example 1: If the area of a sector is $100$ sq. cm and the area of the enclosed triangle is $78$ sq. cm, what is the area of the segment?

Area of the sector = $100$ sq. cm

Area of enclosed triangle = $78$ sq. cm

Area of segment = Area of the sector – Area of an enclosed triangle.

Therefore area of segment = $100 – 78 = 22$ sq. cm

Example 2: Find the area of the major segment of a circle if the area of the corresponding minor segment is $14$ sq. cm and the radius is $7$ cm. Use $\pi = \frac{22}{7}$

Area of minor segment = $14$ sq. cm

Radius of circle = $7$ cm

Area of circle = $\pi r^2 = \frac{22}{7} \times 7^2 = 154 \text{ cm}^2$

Area of major segment = Area of circle – Area of corresponding minor segment

= $154 – 14 = 140 \text{ cm}^2$

Example 3: What is the area of the segment corresponding to the arc subtending an angle of $60^{\circ}$ at the centre of a circle with radius $6$ cm?

Measure of central angle $\theta = 60^{\circ}$

Radius of circle $r = 6$ cm

Area of segment is given by $r^2 \left(\frac{\theta \pi}{360} – \frac{\sin \theta}{2} \right)$

$6^2 \left(\frac{60 \pi}{360} – \frac{\sin 60}{2} \right)$

$= 36 \left(\frac{60 \times \frac{22}{7}}{360} – \frac{\frac{\sqrt{3}}{2}}{2} \right)$

$= 36 \left(\frac{\frac{22}{7}}{6} – \frac{\sqrt{3}}{4} \right)$

$= 36 \left(\frac{22}{42} – \frac{\sqrt{3}}{4} \right)$

$= 36 \left(\frac{11}{21} – \frac{\sqrt{3}}{4} \right)$

$= 36 \left(\frac{11}{21} – \frac{\sqrt{3}}{4} \right)$ sq cm

Perimeter of Segment of a Circle

Consider a circle with centre $\text{O}$ and radius $r$. Further consider two points $\text{A}$ and $\text{B}$ on the circumference of a circle, such that $\text{APB}$ is a chord and $\angle \text{AOB} = \theta$ is a central angle.

Perimeter of segment pf circle = Length of chord $\text{AB}$ + Length of arc $\text{APB}$ 

Length of chord $\text{AB} = 2 r \sin \frac{\theta}{2}$

And length of arc $\text{APB} = \frac{\pi r \theta}{180}$  

Thus perimeter of segment of circle = $2 r \sin \frac{\theta}{2} + \frac{\pi r \theta}{180}$

$= 2r(\sin \frac{\theta}{2} + \frac{\pi \theta}{360})$

Examples on Perimeter of Segment of a Circle

Example 1: Find the perimeter of a segment of a circle of radius $5$ cm and central angle $90^{\circ}$.

Radius of circle $r = 5 $ cm

Central angle $\theta = 90^{\circ}$

Perimeter of segment = $= 2r(\sin \frac{\theta}{2} + \frac{\pi \theta}{360})$

$= 2 \times 5 \left(\sin \frac{90^{\circ}}{2} + \frac{\frac{22}{7} \times 60}{360} \right)$

$= 10 \left(\sin 45^{\circ} + \frac{\frac{22}{7}}{6} \right)$

$= 10 \left(\frac{1}{\sqrt{2}} + \frac{22}{42} \right)$

$= 10 \left(\frac{1}{\sqrt{2}} + \frac{11}{21} \right)$ cm

Practice Problems

Find the perimeter and area of a segment of a circle with

• radius $5$ cm and central angle $45^{\circ}$
• radius $2$ cm and central angle $\frac{\pi}{3}$
• radius $7$ cm and central angle $120^{\circ}$
• radius $10$ cm and central angle $\frac{2 \pi}{3}^{\circ}$

FAQs

Conclusion

A segment of a circle is a region bounded by a chord and a corresponding arc lying between the chord’s endpoints. The perimeter of a segment is calculated using the formula $2r \sin \frac{\theta}{2} + r \theta$ and the area is calculated using the formula $r^2 \left(\frac{\theta \pi}{360} – \frac{\sin \theta}{2} \right)$, where $r$ is the radius of the circle and $\theta$ is the angle subtended by the segment at the centre of the circle.

Recommended Reading

• How to Construct a Tangent to a Circle(With Steps & Pictures)
• Angles in a Circle – Meaning, Properties & Examples
• Tangent of a Circle – Meaning, Properties, Examples
• How to Draw a Circle(With Steps & Pictures)
• Chord of a Circle – Definition, Properties & Examples
• What is a Circle – Parts, Properties & Examples
• Area of a Circle – Formula, Derivation & Examples
• Circumference (Perimeter) of a Circle – Definition, Formula & Examples
• What is an Arc of a Circle?(Definition, Formulas & Examples)
• What is a Sector of a Circle?(Definition, Formulas & Examples)