Working out the area of a segment enclosed in a circle (angle in degrees)

Area Of A Segment Video

The area of a segment can be found by the following mathematical formula:

A = (Ѳ/360)πr²-1/2r²sinѲ

Ѳ is the angle of the sector and r is the radius of the sector.

Basically, (Ѳ/360)πr² gives the area of the sector and 1/2r²sinѲ gives the area of the triangle.

Let’s take a look at an example:

Example 1

Work out the area of the segment shaded in the diagram to the right..

First of all write down the values for Ѳ and the r.

Ѳ = 64⁰ (since this is the angle of the sector)

r = 6cm (since this is the radius of the circle)

All you need to do now is substitute these two values into the formula above to give you the area of the segment.

A = (Ѳ/360)πr²-1/2r²sinѲ

A = (64/360) × π × 6² - ½ × 6² × sin 64

If you have an up-to-date scientific calculator then this can be typed in exactly how it appears on the line above.

This gives the area of the segment as 3.93 cm² rounded to 3 significant figures.

Example 2

Work out the area of a segment if the angle in the sector is 137⁰ and the radius of the sector is 11cm.

Here:

Ѳ = 137⁰ (since this is the angle of the sector)

r = 11 inches (since this is the radius of the circle)

Again, plug these two values into the formula to give you the area of the segment.

A = (Ѳ/360)πr²-1/2r²sinѲ

A = (137/360) × π × 11² - ½ × 11² × sin 137

Like example 1, type this in onto your scientific calculator.

This gives the area of the segment as 144.7 inches² rounded to 3 significant figures.

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