What Is the Area of a Circular Sector?

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Here you will learn about circular sectors and learn how to calculate how many degrees a circular sector has. You can combine this with the area of a circle and have what you need to calculate the area of a circular sector.

You already know that there are 360° in a circle. A circular sector is a portion of an entire circle. It is expressed using an angle, v.

Formula

Area of a Circular Sector Expressed by Angle v

A = v 360° π r2

where v is the angle, which tells you how large the part of circle’s area you’re calculating, is.

Example 1

What is the area of a circular sector at 60° with a radius of 3cm?

We put v = 60° and r = 3 into the formula and calculate:

A = 60° 360° π 32 1 6 3.14 9 = 4.71cm2

Example 2

What is the area of a circular sector at 50° with a radius of 4cm?

We put v = 50 and r = 4 into the formula and calculate:

A = 50° 360° π 42 5 36 3.14 16 6.98cm2

A part of a circle can also be expressed by a circle arc. If this arc length is known, you can calculate the area in this way:

Formula

Area of Circular Sector Expressed by Circular Arc

A = b r 2 ,

where b is the arc length.

The arc length b is part of the circumference, which is 2πr. By multiplying this part of the circumference by the formula of area, you get the area of a circular sector expressed by an arc length:

A = b 2 π r π r2 = b r 2

Example 3

What is the area of a circular sector with circular arc of 6cm and a radius of 5cm?

We put in b = 6 and r = 5 into the formula and calculate:

A = 6 2 π 5 π 52 = 6 5 2 = 15cm2

Example 4

What is the area of a circular sector with circular arc of 2cm and a radius of 7cm?

We put in b = 2 and r = 7 into the formula and calculate:

A = 2 7 2 = 7cm2

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