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.

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`.

What is the area of the sector, shown below?

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 cm^{2}

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

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

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