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Area of Sector

The area of sector of a circle is the space enclosed within the boundary of the sector. A sector always originates from the center of the circle. Let us learn more about the area of sector formula, and how to find the area of a sector in radians and degrees.

What is Area of Sector of a Circle?

The space enclosed by the sector of a circle is called the area of the sector. For example, a pizza slice is an example of a sector that represents a fraction of a pizza. There are two types of sectors: minor and major sectors. A minor sector is a sector that is less than a semi-circle, whereas, a major sector is a sector greater than a semi-circle.

The figure given below represents the sectors in a circle. The shaded region shows the area of the sector OAPB. Here, ∠AOB is the angle of the sector. It should be noted that the unshaded region is also a sector of the circle. So, the shaded region is the area of the minor sector and the unshaded region is the area of the major sector.

Now, let us learn about the area of a sector formula and its derivation after learning about the definition of a sector.

Sector Definition

The sector of a circle is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. It is considered as a portion of a circle with two radii and an arc. A circle is divided into two sectors, the minor sector and the major sector where the minor sector is the smaller portion in the circle and the larger sector is the major portion. The semi-circle is the most common sector of a circle, which represents half a circle.

Area of Sector Formula

In order to find the total space enclosed by the sector, we use the area of a sector formula. The area of a sector can be calculated using the following formulas,

Area of a Sector of Circle = (θ/360º) × πr², where, θ is the sector angle subtended by the arc at the center, in degrees, and ‘r’ is the radius of the circle.
Area of a Sector of Circle = 1/2 × r²θ, where, θ is the sector angle subtended by the arc at the center, in radians, and ‘r’ is the radius of the circle.

Area of Sector Formula Derivation

Let us apply the unitary method to derive the formula for the area of the sector of a circle. We know that a complete circle measures 360º. The area of a circle with an angle measuring 360º at the center is given by πr², where ‘r’ is the radius of the circle.

If the angle at the center of the circle is 1º, the area of the sector is πr²/360º. So, if the angle at the center is θ, the area of the sector is, Area of a Sector of Circle = (θ/360º) × πr², where,

θ is the angle subtended at the center, given in degrees.
r is the radius of the circle.

In other words, πr²represents the area of a full circle and θ/360º tells us how much of the circle is covered by the sector.

If the angle at the center is θ in radians, then the formula for the area of the sector of a circle = (1/2) × r²θ, where,

θ is the angle subtended at the center, given in radians.
r is the radius of the circle.

It should be noted that semi-circles and quadrants are special types of sectors of a circle with angles of 180° and 90° respectively.

Area of Sector in Degrees

Let us use these formulas and learn how to calculate the area of the sector of a circle when the subtended angle is given in degrees with the help of an example.

Example: A circle is divided into 3 sectors and the central angles made by the radius are 160°, 100°, and 100° respectively. Find the area of all the three sectors.

Solution:

The angle made by the first sector is θ = 160°. Therefore, the area of the first sector = (θ/360°) × πr² = (160°/360°) × (22/7) × 6² = 4/9 × 22/7 × 36 = 352/7 = 50.28 square units.

The angle made by the second sector is θ = 100°. Therefore, the area of the second sector is = (θ/360°) × πr² = (100°/360°) × (22/7) × 6² = 5/18 × 22/7 × 36 = 220/7 = 31.43 square units.

The angle made by the third sector is the same as that of the second sector (θ = 100°). Thus, the area of the second sector is equal to the area of the third sector. Therefore, the area of the third sector = 31.43 square units.

Area of Sector in Radians

If we need to find the area of sector when the angle is given in radians, we use the formula, Area of sector = (1/2) × r²θ; where θ is the angle subtended at the center, given in radians, and ‘r’ is the radius of the circle. So, let us understand where the formula comes from. We know that the formula for the area of a sector (in degrees) = (θ/360º) × πr² because it is a fraction of a circle. The same concept is applied to the formula when we want to express it in radians, but we just need to replace 360° with 2π because 2π (in radians) = 360°. This means, Area of sector in radians = (θ/2π) × πr². On further simplifying the sector area formula, we get, area of sector = (θ/2) × r² or (1/2) × r²θ. Let us understand how to find the area of a sector in radians with an example.

Example: Find the area of a sector if the radius of the circle is 6 units, and the angle subtended at the center = 2π/3

Solution: Given, radius = 6 units; Angle measure (θ)= 2π/3

The area of the given sector can be calculated with the formula, Area of sector (in radians) = (θ/2) × r². On substituting the values in the formula, we get Area of sector (in radians) = [2π/(3×2)] × 6² = (π/3) × 36 = 12π.

Therefore, the area of the given sector in radians is expressed as 12π square units.

Real-Life Example of Area of Sector of Circle

One of the most common real-life examples of the area of a sector is the slice of a pizza. The shape of the slices of a circular pizza is like a sector. Observe the figure given below that shows a pizza that is sectioned into 6 equal slices, where each slice is a sector, and the radius of the pizza is 7 inches. Now, let us find the area of the sector formed by each slice by using the sector area formula. It should be noted that since the pizza is divided into 6 equal slices, the angle of sector is 60°. Area of Pizza slice = (θ/360°) × πr² = (60°/360°) × (22/7) × 7² = 1/6 × 22 × 7 = 77/3 = 25.67 square units.

Tips on Area of Sector

Here is a list of a few important points that are helpful in solving the area of sector problems.

The area of a sector of a circle is the fractional area of the circle.
The area of a sector of a circle with radius ‘r’ is calculated with the formula, Area of a sector = (θ/360º) × π r²
The arc length of the sector of radius r can be calculated with the formula, Arc Length of a Sector = r × θ

Perimeter of a Sector

The perimeter of the sector of a circle is the length of two radii along with the arc that makes the sector. In the following diagram, a sector is shown in yellow colour.

The perimeter should be calculated by doubling the radius and then adding it to the length of the arc.

Perimeter of a Sector Formula

The formula for the perimeter of the sector of a circle is given below :

Perimeter of sector = radius + radius + arc length

Perimeter of sector = 2 radius + arc length

Arc length is calculated using the relation :

Arc length = l = (θ/360) × 2πr

Therefore,

Perimeter of a Sector = 2 Radius + ((θ/360) × 2πr )

Example

A circular arc whose radius is 12 cm, makes an angle of 30° at the centre. Find the perimeter of the sector formed. Use π = 3.14.

Solution :

Given that r = 12 cm,

θ = 30° = 30° × (π/180°) = π/6

Perimeter of sector is given by the formula;

P = 2 r + r θ

P = 2 (12) + 12 ( π/6)

P = 24 + 2 π

P = 24 + 6.28 = 30.28

Hence, Perimeter of sector is 30.28 cm
Practice Questions

A sector is cut from a circle of radius 21 cm. The angle of the sector is 150^o. Find the length of the arc, perimeter and area of the sector.
A pizza with 21 cm radius is divided into 6 equal slices (slices are in the shape of a sector). Find the area of each slice.
The minute hand of a clock is 7 cm long. Find the area swept by the minute hand in 35 minutes.

Area of a Sector Examples

Example 1: If the angle of a sector of a circle is 60°, and the radius of the circle is 7 inches, what is the area of the sector of this circle?

Solution:The radius of the circle is 7 inches and the angle is 60°. So, let us use the area of sector formula. The area of sector = (θ/360°) × π r² = (60°/360°) × (22/7) × 7² = 77/3 = 25.67 square units. Therefore, the area of the minor sector is 25.67 square units.

Example 2: An umbrella has equally spaced 8 ribs. If viewed as a flat circle of radius 7 units, what would be the area between two consecutive ribs of the umbrella? (Hint: The area between two consecutive ribs would form a sector of a circle)

Solution:The radius of the flat umbrella = 7 units. There are 8 ribs in the umbrella. Since a complete angle of a circle = 360°, the angle of each sector of the umbrella = 360/8 = 45° because the circle is divided into 8 equal sectors. Thus, the area of sector = (θ/360°) × π r² = (45°/360°) × 22/7 × 7² = 77/4 = 19.25 square units. Therefore, the area between two consecutive ribs of the umbrella is 19.25 square units.

Example 3: A circle with a diameter of 2 units is divided into 10 equal sectors. Can you find the area of each sector of the circle?

Solution:The diameter of the circle is 2 units, therefore, the radius of the circle is 1 unit. Since a complete angle of a circle = 360°, the angle of each sector of the circle is 360/10 = 36° because the complete angle is divided into 10 equal parts. Area of Sector = (θ/360°) × πr² = 36°/360° × 22/7 × 1 = 11/35 = 0.314 square units. Therefore, the area of each sector of the circle is 0.314 square units.

Example 1: If the angle of the sector with radius 4 units is 45°, then find the length of the sector.

Solution: Area = (θ/360°) × πr²

= (45°/360°) × (22/7) × 4 × 4

= 44/7 square units

The length of the same sector = (θ/360°)× 2πr

l = (45°/360°) × 2 × (22/7) × 4

l = 22/7

Example 2: Find the area of the sector when the radius of the circle is 16 units, and the length of the arc is 5 units.

Solution: If the length of the arc of a circle with radius 16 units is 5 units, the area of the sector corresponding to that arc is;

A = (lr)/2 = (5 × 16)/2 = 40 square units.

FAQs on Area of Sector of Circle

What is the Area of a Sector of a Circle?

The space enclosed by the sector of a circle is called the area of the sector of a circle. The part of the circle that is enclosed by two radii and the corresponding arc is called the sector of the circle.

What is the Formula for Area of Sector of Circle?

The two main formulas that are used to find the area of a sector are:

Area of a Sector of Circle = (θ/360º) × πr², where, θ is the angle subtended at the center, given in degrees, and ‘r’ is the radius of the circle.
Area of a Sector of Circle = 1/2 × r²θ, where, θ is the angle subtended at the center, given in radians, and ‘r’ is the radius of the circle.

How to Calculate Area of Sector using Degrees?

When the angle subtended at the center is given in degrees, the area of a sector can be calculated using the following formula, area of a sector of circle = (θ/360º) × πr², where, θ is the angle subtended at the center, given in degrees, and r is the radius of the circle.

What do you Mean by Sector of a Circle?

A sector is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. The semicircle is the most common sector of a circle, which represents half of a circle.

What do you Mean by the Arc of a Circle?

A part of a curve or a part of a circumference of a circle is called the arc. Many objects have a curve in their shape. The curved portion of these objects is mathematically referred to as an arc.

How is the Area of Sector of Circle Formula Derived?

The area of the sector shows the area of a part of the circle’s area. We know that the area of a circle is calculated with the formula, πr². The formula for the area of a sector of a circle is derived in the following way:

Apply the unitary method to derive the formula of the area of a sector of circle.
We know, a complete circle measures 360º. The area of a circle with an angle measuring 360º at the center is given by πr², where r is the radius of the circle.
If the angle at the center of the circle is 1º, the area of the sector is πr²/360º. So, if the angle at the center is θ, the area of the sector is, Area of a Sector of a Circle = (θ/360º) × πr², where, θ is the angle subtended at the center, given in degrees, and r is the radius of the circle.
In other words, πr²represents the area of a full circle and θ/360º tells us how much of the circle is covered by the sector.

How to Find the Area of Sector with Arc Length and Radius?

The area of a sector can be calculated if the arc length and radius is given. We first calculate the angle (θ) subtended by the arc with the formula, Length of Arc = (θ/360) × 2πr. Now, we already know the radius, and once the angle is known, the area of the sector can be calculated with the formula, Area of a Sector of a Circle = (θ/360º) × πr²

How to Find the Radius from Area of Sector?

If the area of a sector is known, and the angle (θ) subtended by the arc is known, the radius can be calculated by substituting the given values in the formula, Area of a Sector of a Circle = (θ/360º) × πr². For example, let us find the radius if the area of a sector is 36π, and the sector angle is given as 90°. We will substitute the given values in the formula, Area of a Sector of a Circle = (θ/360º) × πr², that is, 36π = (90/360) × πr². So, the value of r² = 144, which means r = 12 units.

How to Find the Area of Sector in Terms of Pi?

The area of sector can also be expressed in terms of pi (π). For example, if the radius of a circle is given as 4 units, and the angle subtended by the arc for the sector is 90°, let us find the area of the sector in terms of pi. Area of sector = (θ/360º) × πr². Substituting the values in the formula, Area of sector = (90/360) × π × 4². After solving this, we get, the area as 4π.

How to Find the Area of a Sector in Radians?

In order to find the area of a sector with the central angle in radians, we use the formula, Area of sector = (θ/2) × r²; where θ is the angle subtended at the center, given in radians, and ‘r’ is the radius of the circle. For example, if the radius of the circle is 12 units, and the sector angle subtended by the arc at the center = 4π/3, let us find the area of the sector. Area of sector (in radians) = (θ/2) × r². On substituting the values in the formula, we get Area of sector (in radians) = [4π/(3×2)] × 12² = (2π/3) × 144 = 96π. Therefore, the area of the sector in radians is expressed as 96π square units.

How to Find the Area of a Sector Without Angle?

If the sector angle is not given, but we know the arc length and the radius, the area of a sector can be calculated. We first find the sector angle by substituting the given values of the arc length and radius in the formula, Length of Arc = (θ/360) × 2πr. After calculating the angle, we can easily find the area of the sector with the formula, Area of a Sector of a Circle = (θ/360º) × πr².

How to Find the Arc Length of a Sector?

Arc length is the distance along the part of the circumference of a circle. The arc length of a circle can be calculated using the following formulas:

Arc Length = θ × r; where θ = Central angle subtended by the arc, and r = radius of the circle. This formula is used when θ is in radians.
Arc Length = θ × (π/180) × r; where θ = Central angle subtended by the arc, and r = radius of the circle. This formula is used when θ is in degrees.

What is meant by the sector of a circle?

The sector of a circle is the region bounded by two radii and an arc of a circle.

What is the formula for the arc length of the sector of a circle?

Let PQ is an arc of a circle of radius r and centre at O if PQ subtends angle 𝜃 at the centre of the circle. Then, the arc length of PQ = 𝜃/360^o (2𝜋r), where 𝜃 is measured in degrees.

How to Find the Area of Minor Sector?

The area of minor sector is the area of the sector which is smaller than half of that particular circle. We know that a minor sector is a sector that is less than a semi-circle, whereas, a major sector is a sector greater than a semi-circle. The formula which is used to find the minor sector is the same which is used to find the area of any sector of a circle. This means the area of a minor sector can be calculated using the following formulas,

Area of Sector of Circle = (θ/360º) × πr², where, θ is the angle subtended at the center, given in degrees, and ‘r’ is the radius of the circle.
Area of Sector of Circle = 1/2 × r²θ, where, θ is the angle subtended at the center, given in radians, and ‘r’ is the radius of the circle.

How to Find Radius with Arc Length and Sector Area?

The radius of a circle can be calculated if we know the arc length and the sector area. A simple way is to substitute the given values in the formula, Area of sector = (Arc length × radius)/2. Let us understand this with an example. For example, if the arc length of a circle is given as 15 cm and the area of the sector is 225 cm². We know that, Area of sector = (Arc length × radius)/2. After substituting the values in this formula, we get, 225 = (15 × radius)/2. This can be further solved as, (225 × 2)/15 = radius. We get the radius as 30 cm.
This formula has been derived from the relationship, Area of sector = Area of circle × ( Arc length/Circumference of circle). This can be further written as, Area of sector = πr² × ( Arc length/2πr). So, after further simplifying this formula, we get, Area of sector = (Arc length × r)/2

What is a Circle?

A circle is a locus of points equidistant from a given point located at the centre of the circle. The common distance from the centre of the circle to its point is called the radius. Thus, the circle is defined by its centre (o) and radius (r). A circle is also defined by two of its properties, such as area and perimeter. The formulas for both the measures of the circle are given by;

Area of a circle = πr²The perimeter of a circle = 2πr

What is Sector of a circle?

The sector is basically a portion of a circle which could be defined based on these three points mentioned below:

A circular sector is the portion of a disk enclosed by two radii and an arc.
A sector divides the circle into two regions, namely Major and Minor Sector.
The smaller area is known as the Minor Sector, whereas the region having a greater area is known as Major Sector.