Area of a Sector of a Circle Formula: Derivation & Examples

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Sector of a Circle is a section of a circle formed or enclosed by two radii and an arc of a circle. A sector of a circle is a pie-shaped section of a circle bounded by the two radii of the circle and the arc that connects them. A semicircle, which is the half of a circle, is the most common sector of a circle. The radius of a circle is a straight line segment from the center of a circle to the circumference of a circle. An arc is defined as a part of a curve lying on the circumference of a circle.

Area of a Sector of a Circle is the space enclosed inside a sector of a circle. The Area of a Sector Formula is A = (θ/360°) × πr², where θ is the sector angle subtended by the arcs at the center and r is the radius. The length of the arc of a sector of a circle is calculated using the formula (θ/360°) × 2πr.

Table of Content

What is Sector?
Length of Arc of a Sector
Area of a Sector Formula
Derivation of Area of a Sector Formula
Solved Examples on Area of a Sector Formula
Things to Remember
Sample Questions

Key Terms: Sector, Area of a Sector, Arc, Radius, Circle, Minor Sector, Major Sector, Circumference, Angle, Area of a Sector Formula

What is Sector?

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A sector is a portion of a circle enclosed by two radii and an arc. The arc is a section of the circumference of circle. A sector of a circle is produced when a portion of the circumference (arc) of the circle meets with two radii on the two extreme points of the arc.

A circle can be divided into two parts when a sector is formed inside it:

Major Sector: The larger portion is known as the major sector. Major sectors form an angle of more than 180 degrees.
Minor Sector: The smaller area is known as the minor sector. Minor sectors form an angle of less than 180 degrees.

Sectors of a Circle

Sectors of a Circle

The circle is a perfectly symmetric figure, therefore, all the sectors that are present within the same circle or the congruent circles, which have the same or common central angles will be congruent sectors.

Relevant Topics
Areas Related To Circles Formula	Area of Segment of a Circle	Great Circle Formula
Area of a Circle Formula	Tangent to a Circle	Radius Formula

Length of Arc of a Sector

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The length of an arc of a sector of a circle can be measured by the formula

\({\theta \over 360}\times2\pi r\)

Arc

Arc

Read More: Arc Length Formula

Area of a Sector Formula

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An entire circle has a measurement of 360 degrees. The area of a circle is given by π times the square of the radius length of the circle, or πr². Therefore the area of a sector of a circle with a radius length of r and measurement of the angle of degrees is given by

\({\theta \over 360}\times\pi r^2\)

Derivation of Area of a Sector Formula

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Let us take a circle with the center O and a radius length of r. Now, we will take a sector of the circle with an angle subtended at the center of measurement of θ degrees.

Sector of a Circle

Now, we know that the area of this complete circular region is πr². This region is a sector in itself of measurement of 360 degrees.

Therefore, by using the unitary method, the area of the particular sector can be calculated as:

Area of the region when the angle at the center is of 360 degrees = πr²
Area of the region when the angle at the center forms a measurement of θ = \({\theta \over 360}\times\pi r^2\).

In conclusion, when the angle is θ, the area of a sector of a circle is \({\theta \over 360}\times\pi r^2\).

Solved Examples on Area of a Sector

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Example 1: Calculate the area of the sector for a given circle of radius 5 cm if the angle of its sector is 30°.

Solution:It is given that,

r = 5
θ = 30°

Use the formula Area of a Sector Formula,

A = (θ/360°) × πr²

A = (30/360) × (22/7) × 5²

= 550/840

= 0.65 sq. cm

Thus, the area of the sector of the given circle is 0.65 sq. cm.

Example 2: What will be the area of the sector for a circle of radius 9 cm if the angle of its sector is 45°.

Solution: Given that,

r = 9
θ = 45°

Using the formula Area of a Sector Formula,

A = (θ/360°) × πr²

A = (45/360) × (22/7) × 9²

= 1782/56 = 31.82 sq. cm

Hence, the area of the sector is 31.82 sq. cm.

Example 3: Calculate the value of the angle subtended at the center of the circle if the area of its sector is 770 sq. cm and its radius is 7 cm.

Solution: Given to us is that

r = 7
Area = 770 sq. cm

We will use the Area of a Sector Formula,

A = (θ/360°) × πr²

Substituting the values,

770 = (θ/360) × (22/7) × 7²

770 = (θ/360) × 154

θ/360 = 5

θ = 1800°

Thus, the value of the angle subtended at the center of the circle is 1800°.

Related Topics
Concentric Circles	Tangent Circle Formula	Sphere Formula
Central Angle of Circle	Standard Equation of a Circle	Chord of Circle Length Theorem

Things to Remember

A sector is a portion of a circle enclosed by two radii and an arc.
The formula for calculating the area of a sector of a circle can be derived from the formula of the area of a circle.
Area of a Sector of a Circle = \({\theta \over 360}\times\pi r^2\).
An arc is a section of the circumference of a circle.
Length of Arc of a Sector of a Circle = \({\theta \over 360}\times2\pi r\).
Minor sectors form an angle of less than 180 degrees
major sectors form an angle of more than 180 degrees.
A semicircle, which is the half of a circle, is the most common sector of a circle.

Sample Questions

Ques. Find the area of the sector for a given circle of radius 4 units, if the angle of its sector is 45°. (3 Marks)

Ans. It is given that,

Radius r = 4 units
Angle θ = 45°

Using the formula Area of a Sector Formula,

A = (θ/360°) × πr²

= (450/3600)×22/7×4²

= 6.28 sq. units

Thus, the area of the sector of the given circle is 6.28 sq. units.

Ques. What will be the area of the sector with a central angle of 30° and a radius of 9 cm? (3 Marks)

Ans. Given that,

Radius r = 9 cm
Angle θ = 30°

Using the formula Area of a Sector Formula,

A = (θ/360°) × πr²

(300/3600)×22/7×9²

= 21.21 cm²

Hence, the area of the sector is 21.21 cm².

Ques. For a given circle of radius 5 units, the angle of its sector is 50°. Find the area of the sector. (3 Marks)

Ans. Given,

Radius r = 5 units
Angle θ = 50°

Therefore, Area of the sector = \({\theta \over 360}\times\pi r^2\)

= 50/360 x 22/7 x 5 x 5

= 27500/ 2520 = 1.09 unit squares.

Ques. Find the perimeter and the area of a sector of a circle of radius 40 cm whose central angle is 90°. (3 Marks)

Ans. Given that,

Radius r = 40 cm
Angle = 90 degrees

Length l = \({\theta \over 360}\times2\pi r\)= 90/360 x 2 x 22/7 x 40 = 62.85 cm

Therefore, Perimeter = l + 2r = 62.85 + 2 x 40 = 142.85 cm

Area of the sector of the circle = ½ x lr = ½ x 62.85 x 40 = 1257 cm square.

Ques. If the difference between the circumference and the radius of a circle is 37 cm, then using π = 22/7, calculate the circumference (in cm) of the circle. (3 Marks)

Ans. Given, circumference of the circle - radius of the circle = 37 cm

This implies, 2πr - r = 37 cm

⇒ r(2π – 1) = 37

⇒r[44/7 - 1] = 37

⇒r(37/7)= 37

⇒r = 37*7 / 37

⇒ r = 7 cm

Therefore, circumference = 2πr = 2.22/7 . 7 = 44 cm.

Therefore, the circumference of the circle is 44 cm and the radius of the circle is 7 cm.

Ques. If π is taken as 22/7, calculate the distance (in metres) covered by a wheel of diameter 35 cm, in one revolution. (3 Marks)

Ans. It is given that Radius (r) = 35/2

Distance (in metres) covered by a wheel = Perimeter = 2πr

= 2* 22/7 * 35/2

= 110 cm or 1.1 m

Therefore, the distance (in metres) covered by a wheel of diameter 35 cm, in one revolution is 110 cm or 1.1 metres.

Ques. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. (3 Marks)

Ans. The area formed by the minute hand when it is 5 minutes is the shape of a sector of a circle.

Therefore, the area swept by the minute hand = area of a sector of a circle =\({\theta \over 360}\times\pi r^2\) .

In this case, θ = 360/60 minutes x 5 minutes = 30 degrees. [Since 1 hour = 60 minutes, using unitary method]

Radius (r) = 14 cm

Therefore, area of the sector = \({\theta \over 360}\times\pi r^2\)

= 30/360 x 22/7 x 14 x 14 = 154/3 cm square = 51.3333 cm square.

Ques. Two circular pieces of equal radii and maximum area, touching each other are cut out from rectangular cardboard of dimensions 14 cm × 7 cm. Find the area of the remaining cardboard. (3 Marks)

Ans. Here radius =7/2 cm, Length = 14 cm, Breadth = 7 cm

Area of the remaining cardboard = area of(rectangle) – 2(area of circle)

= L x B – 2πr²

=14 × 7 – 2 x 22/7 x 7/2 x 7/2

= 98 – 77 = 21 cm square

Therefore, the area of the remaining cardboard is 21 cm square.

Ques. Area of a sector of a circle of radius 14 cm is 154 cm². Find the length of the corresponding arc of the sector. (3 Marks)

Ans. Area of sector = 154 cm square

L x r x ½ = 154

⇒ ½ x l x 14 = 154

⇒7l = 154

⇒ l = 22 cm

Therefore, the length of the corresponding arc is 22 cm.

Ques. From a rectangular sheet of paper ABCD with AB = 40 cm and AD = 28 cm, a semi-circular portion with BC as diameter is cut off. Find the area of the remaining paper. (3 Marks)

Ans. Given that,

Length of paper (l) AB = 40 cm,
Width of paper (b) AD = 28 cm,

Area of paper = l × b = 40 × 28 = 1120 sq. cm
Diameter of the semi-circle = 28 cm
Radius of semi-circle, r = 14 cm

Area of semicircle = ½ πr²= ½ x 22/7 x 14 x 14 = 308 cm square.

Therefore, the Area of remaining paper = 1120 – 308 = 812 square cm.

Ques. If the circumference and the area of a circle are numerically equal, find the diameter of the circle. (2 Marks)

Ans. Here, we are given, 2πr = πr²

⇒ 2r = r²

⇒ r(r – 2) = 0 or r = 2

Diameter of the circle = 2 x radius = 2r = 2x 2.

Therefore, the diameter is 4 units.

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Area of a Sector of a Circle Formula: Derivation & Examples

What is Sector?

Length of Arc of a Sector

Area of a Sector Formula

Derivation of Area of a Sector Formula

Solved Examples on Area of a Sector

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

4.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

5.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

Similar Mathematics Concepts

Comments

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Area of a Sector of a Circle Formula: Derivation & Examples

What is Sector?

Length of Arc of a Sector

Area of a Sector Formula

Derivation of Area of a Sector Formula

Solved Examples on Area of a Sector

Things to Remember

Sample Questions

CBSE X Related Questions

1.Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

4.Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

5.Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent. Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Two dice are rolled together. The probability of getting an outcome $(x, y)$ where $x \gt y$, is

2.
In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10\text{ cm}$, $EB = CF = 5\text{ cm}$ and $AB = 7\text{ cm}$, then find the length $DE$.

3.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

4.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

5.
Assertion (A) : The system of linear equations $3x - 5y + 7 = 0$ and $-6x + 10y + 14 = 0$ is inconsistent.
Reason (R) : When two linear equations don't have unique solution, they always represent parallel lines.

6.
Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.