How to Calculate Area of Sector of Circle?

Area of the sector is easily calculated by using various formulas of geometry. In this article, we have covered a definition of sector circles, types of sectors, and others in detail.

Sector Definition

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. The image of the sector of the circle is shown below:

Sector Definition

Properties of Sector

• A sector of a circle always originates from the centre of the circle.
• Semi-circle is the most common sector of a circle, with the angle between the radii equal to 180°.
• Area of 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,

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

Examples on Area of Sector of Circle

Example 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°

Radius of the circle = 5 units

Thus, aea 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.

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

Area of the sector = 1/2 × r² × θ

Given, radius of circle is 8cm. Thus,

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

Approximating Value of π = 3.14, we get,

Area of sector = 16 × 3.14 = 50.24 cm²

Example 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:

Area of the circle is 50cm².

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

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

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

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

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

So, 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

FAQs on Area of Sector of Circle

What is a Circle?

A circle is curve in which the locus of the curve is always equidistant form a fixed point(center of circle).

What is Sector of Circle?

Sector of circle is defined as the part of circle which is formed by joining two radius of the circle.

What is Formula for Area of Sector Circle?

Formula for area of sector of a circle is (θ/360°) × πr². Where r is radius of circle and 𝜃 is angle of sector.