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Circles - Arcs Sectors

Circles - Arcs Sectors

An arc is a fraction of the circumference of a circle.

The amount of the circumference that is the arc is given by the number of degrees at the centre.

Arc and sector of a circle

The circumference of the whole circle is given by `2pir`, and the angle at the centre is 360º.

If the angle of the arc is 83º, then the fraction is `frac(83)(360)` of a whole circle:

The arc length is therefore `frac(83)(360) xx 2pir`.

Note that if the perimeter of the sector is required, then include the two radii: `frac(83)(360) xx 2pir + 2r`.

The area of a whole circle is `pir^2`. The area of a sector uses the same fraction: `frac(83)(360) xx pir^2`.

Example 1

What is the area of the sector, shown below?

Sector of a circle question

Area of a circle A `= pir^2`
Area of this sector A `= frac(70)(360) xx pir^2`
Substitute A `= frac(70)(360) xx pi xx 12^2`
`= 87.96`

Answer: 87.9 cm2

Example 2

What is the length of the perimeter of the sector shown below? Give your answer to 1 decimal place.

Sector of a circle question

Perimeter of a circle P `= 2pir`
Perimeter of a sector P `= frac(70)(360) xx 2pir + 2r`
Substitute P `= frac(70)(360) xx 2pi(12) + 2(12)`
`= 14.65 + 24`
`= 38.66`
round to 1dp `= 38.7`

Answer: 38.7 cm