How to calculate the Area of a Sector of Circle?

Sector is defined as the portion inside a circle bounded between two radii and the arc of the circle. The area of this sector depends upon the corresponding angle between the two radii. 

Properties:

• A sector of a circle always originates from the center of the circle.
• The semi-circle is the most common sector of a circle, with the angle between the radii equal to 180°.
• The area of the semi-circle depends upon the radii of the circle.

Types of Sectors

There are three types of sectors depending upon the angle between the corresponding two radii of the sector. They are:

• Minor sector: When the angle enclosed between the two radii is less than 180°. The area enclosed is also smaller than a semi-circle.
• Semi-circle: When the angle enclosed between the two radii is equal to 180°.
• Major sector: When the angle enclosed between the two radii is greater than 180°. The area enclosed is also greater than a semi-circle.

Formula for Area of a Sector

For a circle having radius equals to ‘r’ units and angle of the sector is θ (in degrees), the area is given by,

A circle with radius r 

Area of sector = θ / 360° × πr² 

When θ is given in radian, the area is given by

Area of sector = 1/2 × r²θ

Proof:

For a circle with radius r units, the area is given by πr².

Now the fraction of the area enclosed by the sector will be the same as the fraction of the angle enclosed by the sector in the circle.

Thus, the fraction of area enclosed = θ / 360°

So, the area enclosed by the sector = (θ / 360°) × πr²

Sample Problems

Question 1. Find the area of the sector of a circle whose angle enclosed equals 60^o and the radius of the circle is 5 units. It is a major or minor sector?

Solution:

Give, the angle of the sector = θ = 60°

The radius of the circle = 5 units

Thus, the area of the sector = 60°/360° × π × 5² = 25π/6

Approximating the value of π = 3.14, we get,

Area of sector = 25 × 3.14 / 6 = 13.08 sq. units

Since, angle of sector is less than 180°, it is a minor sector.

Question 2. Find the area of a sector whose angle is given as π/2 radians and the radii of the circle is 8cm.

Solution:

Since the angle of the sector is given in radian, we can write,

Area of the sector = 1/2 × r² × θ

Given, the radius of the circle is 8cm. Thus,

Area of sector = 1/2 × 8² × π/2 = 16π cm²

Approximating the value of π = 3.14, we get,

Area of sector = 16 × 3.14 = 50.24 cm²

Question 3. For a circle of a given area 50cm², there are three sectors of area 25cm², 45cm², and 13cm². Classify the given sectors among the minor sector, semi-circle, and major sector.

Solution:

The area of the circle is 50cm².

Thus, half of the area of the circle is 50/2 = 25cm². 

Thus, the sector with an area of 25cm² is a semi-circle.

The sector with an area of 45cm² has a greater area than a semi-circle. Thus, it is a major sector.

Lastly, the sector with an area of 13cm² has a smaller area than a semi-circle. Thus, it is a minor sector.

Question 4. If a pizza of radius 5 inches is divided into 6 equal slices, find the area enclosed and angle of each slice of pizza.

Solution:

Since we divide a pizza into 6 equal pieces, each piece represents a sector with an angle equal to one-sixth of the total angle of pizza, that is 360^o.

So, the angle of each pizza slice = 360°/6 = 60°.

So, the area of each sector is given by,

Area of each slice = (θ / 360°) × πr²,

where,

θ = 60°

r = 5 inches

Thus, we get, area of each slice = 60°/360° × π × 5² = 25π/6 sq. inch 

Putting the value of π = 3.14, we get

Area of each slice =  25 × 3.14 / 6 = 13.08 sq. inch