Areas of Sector and Segment of a Circle: Formula, Examples

The area of a sector of a circle is the amount of space covered within the sector’s boundary. A sector always emerges from the centre of the circle. The semi-circle is also a sector of a circle. An arc and a circle chord bound the area of sector and segment of a circle.

When we divide something into parts, each part is referred to as a segment. It is a particular part cut by a chord of the circle. Let’s learn more about the area of a sector and the area of a segment of a circle.

Area of Sector Definition

A sector of a circle is defined as the part of a circle bounded by two radii and the arc connecting them. The space occupied by the sector of a circle is called the area of a sector of a circle.

There exist two types of sectors: major sector and minor sector. A major sector is a sector that is greater than a semi-circle, whereas a minor sector is a sector that is less than a semi-circle. The figure below represents sectors in a circle. The shaded part \(O A P B\) is the area of the minor sector, and the unshaded part \(O A Q B\) is the area of the major sector of the circle.

Example:

One of the most common real-life examples of the area of a sector is a slice of a pizza. The shape of slices of a pizza is similar to a sector of a circle. A pizza of \(7\) inches radius is divided into 6 equal-size slices, as shown in the below figure. Each slice is a sector.

Area of the Sector of a Circle Formula

The area of the sector of a circle formula can be calculated to find the total space covered by part of a circle. The area of a sector can be calculated in two different ways based on the unit of an angle given. The following are the formulas to compute the area of a sector of a circle:

Learn about Area of a Circle

1. Area of a sector of circle \(=\left(\frac{\theta}{360^{\circ}}\right) \times \pi r^{2}\) where \(r\) is the radius of the circle and \(\theta\) (in degrees) is the angle subtended by the arc at the centre.
2. Area of a sector of circle \(=\frac{1}{2} \times r^{2} \theta\) where \(r\) is the radius of the circle and \(\theta\) (in radians) is the angle subtended at the centre.

Segment of a Circle Definition

A segment of a circle is the area that is covered by a chord and an arc of the circle. Let us define a chord and an arc of the circle.

1. A chord is a line segment joining any two points on the circumference of the circle.
2. An arc is a portion of the perimeter (circumference) of a circle.

There are two types of segments. A major segment and a minor segment. A major segment is made by a major arc, and a minor arc makes a minor segment of the circle.

Area of a Segment of Circle

In the figure given below, an arc \(P Q\) and two radii \((O P\) and \(O Q)\) of a circle form a sector. These two radii and the chord of the segment together form a triangle \(P O Q\). Thus, the area of a segment of a circle can be obtained by subtracting the area of the triangle \(P O Q\) from the area of the sector \(P O Q\). i.e.,

The area of a segment of circle \(=\) area of the sector \(-\) area of the triangle

Remember that this is the area of the minor segment. Generally, a segment of a circle means a minor segment. To find the area of the major segment of a circle, we need to subtract the area of the minor segment from the total area of the circle.

Area of a Segment of Circle Formula

Consider the minor segment made by the chord \(P Q\) of the circle of radius \(r\) shown in the below figure. Every arc subtends an angle at the centre called the central angle of the arc. \(\theta\) is the central angle made by the arc \(P Q\). We know that area of the triangle \(O P Q\) from trigonometry is \(\frac{1}{2} r^{2} \sin \theta\).

Also, the area of the sector \(O P Q\) is given by

1. \(\left(\frac{\theta}{360^{\circ}}\right) \times \pi r^{2}\), if \(\theta\) is in degrees
2. \(\frac{1}{2} \times r^{2} \theta\), if \(\theta\) is in radians

Hence, the formulas to compute the area of the minor segment of the circle are:

1. \(\left( {\frac{\theta }{{{{360}^{\rm{o}}}}}} \right) \times \pi {r^2} = \left( {\frac{1}{2}} \right){r^2}\sin \theta \) (OR) \({r^2}\left[ {\pi \left( {\frac{\theta }{{360^{\rm{o}}}}} \right) – \frac{{\sin \,\theta }}{2}} \right],\) when \(\theta\) is given in degrees
2. \(\left( {\frac{1}{2}} \right) \times {r^2}\theta – \left( {\frac{1}{2}} \right){r^2}\sin \,\theta \) (OR) \(\left( {\frac{{{r^2}}}{2}} \right)\left[ {\theta – \sin \,\theta } \right],\) when \(\theta\) is given in radians

Theorems on Segment of a Circle

Majorly, 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 equal always.

Alternate Segment Theorem

The 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 circle’s circumference through the chord’s endpoints.

Learn About Areas Related to Circles

Solved Examples

Q.1. \(PQ\) is a chord of a circle that subtends an angle of \(90^\circ \) at the centre of a circle, and the diameter of the circle is \(14 \,\text {cm}\). Calculate the area of the minor sector of this circle?
Ans: The diameter of the circle \(=14 \mathrm{~cm}\).
The radius of circle \(=\frac{14}{2}=7 \mathrm{~cm}\)
Using the formula of the area of a sector of the circle
The area of minor sector \(=\left(\frac{\theta}{360^{\circ}}\right) \times \pi r^{2}=\left(\frac{90^{\circ}}{360^{\circ}}\right) \times\left(\frac{22}{7}\right) \times 7^{2}=\frac{11 \times 7}{2}=38.5 \mathrm{sq.cm}\).
Therefore, the area of the minor sector is \(38.5 \mathrm{sq} \cdot \mathrm{cm}\).

Q.2. A circle-shaped badge is divided into \(6\) sectors by \(3\) wires. The diameter length is \(2\) units. Work out the area of each sector of the badge?
Ans: Given, the diameter of the badge, \(d\) is \(2\) units.
\(\therefore\) Radius, \(r=\frac{d}{2}=1\) unit.
The angle of each sector of the badge is \(60^{\circ}\) as the complete angle is divided into \(6\) parts.
Area of Sector \(=\left(\frac{\theta}{360^{\circ}}\right) \times \pi r^{2}=\frac{60^{\circ}}{360^{\circ}} \times \frac{22}{7} \times 1=\frac{11}{21}\) sq.units
Therefore, the area of each sector of the badge is \(\frac{11}{21}\) square units.

Q.3. An umbrella has equally spaced \(10\) ribs if viewed as a flat circle of radius \(7\) units. Compute the area between two consecutive ribs of the umbrella.
Ans: The radius of the flat umbrella \(=7 \,\text {units}\)
There are \(10\) ribs in the umbrella.
The central angle of every sector of the umbrella is \(36^{\circ}\) because the complete angle is divided into \(10\) parts.
Thus, the area of sector \(=\left(\frac{\theta}{360^{\circ}}\right) \times \pi r^{2}=\left(\frac{36^{\circ}}{360^{\circ}}\right) \times \frac{22}{7} \times 7^{2}=15.4 \mathrm{sq.units}\).
Therefore, the area between the two adjacent ribs of the umbrella is \(15.4 \,\text {sq.units}\).

Q.4. If the area of a sector is \(80 \,\text {sq.cm}\) and the area of the enclosed triangle is \(68 \,\text {sq.cm}\), calculate the area of the segment?
Ans: Using the relation, area of the segment \(=\) area of the sector \(-\) area of the triangle
\(=(80-68) \,\mathrm{sq} . \mathrm{cm}\)
\(=12 \,\mathrm{sq.cm}\)
Thus, the area of the segment is \(12 \,\mathrm{sq.cm}\).

Q.5. Calculate the area of the major segment of a circle if the area of its minor segment is \(54 \,\text {sq. units}\) and the radius is \(7 \,\text {units}\). Use \(\pi=\frac{22}{7}\).
Ans: Using the relation, area of the major segment \(=\) area of the circle \(-\) the area of the minor segment
\(=\pi r^{2}-54\)
\(=\frac{22}{7} \times 7 \times 7-62\)
\(=92 \,\text {sq.units}\)
Therefore, the area of the major segment is \(92 \,\mathrm{sq. units}\)

Summary 

This article comprises two significant topics: the area of the sector of a circle and the area of the segment of a circle. We learnt the definitions of both. Then we studied major and minor sectors of a circle and major and minor segments of a circle.

Further, we studied the formula to calculate the area of the sector of a circle and the area of a segment of a circle. Lastly, we have briefly discussed the theorems based on the segment of a circle.

FAQs

Q.1. What is the area of the segment of the circle?
Ans: The area of a segment of a circle is the area obtained by subtracting the area of the triangle (formed between two radii and the chord of the segment) from the area of the sector.

Q.2. How do you find the area of a sector of a circle?
Ans: We know that a full circle is \(360^{\circ}\) in measurement. The area of a circle is given as \(\pi\) times the square of its radius length. So if the radius of a circle is \(r\) and the central angle is \(\theta\), then the area of the sector is given by:
area of sector \(=\left(\frac{\theta}{360^{\circ}}\right) \times \pi r^{2}\)

Q.3. What are the steps to follow in finding the area of the segment of the circle?
Ans: Steps to find the area of the segment of a circle:
(i) From the given radius of the circle and measure of the central angle occupied by an arc, determine the area of the sector of the circle. 
(ii) Find the area of the triangle formed between two radii and the chord of the segment.
(iii) Subtract the area of the triangle from the area of the sector to obtain the segment area.

Q.4. What is the difference between a segment of a circle and a sector of a circle?
Ans: A segment of a circle is the region enclosed by a chord and its corresponding arc, while a sector of a circle is the region enclosed by two radii and the corresponding arc.

Q.5. How does the area of a sector relate to the area of a circle?
Ans: If the angle at the centre of the circle is \(1^{\circ}\), the area of the sector is \(\frac{\pi r^{2}}{360^{\circ}}\). If the angle at the centre is \(\theta\), the area of a sector of a circle is \(\left(\frac{\theta}{360^{\circ}}\right) \times \pi r^{2}\) where \(r\) is the radius of the circle and \(\theta\) is the angle subtended at the centre in degrees.