Measurement

Hong Kong

Stage 1 - Stage 3

Lesson

Like we saw in our chapters describing circumferences and areas of circles we now have the following rules.

About circles

$\text{Circumference, C}=2\pi r$Circumference, C=2π`r`

$\text{Area, A}=\pi r^2$Area, A=π`r`2

What if we don't have an entire circle?

Well, half a circle would have half the area or half the circumference. One quarter of a circle would have a quarter of the area, or a quarter of the circumference. In fact all we need to know is what fraction the sector is of a whole circle. For this all we need to know is the angle of the sector.

Looking at the quarter circle, the angle of the sector is $90$90°. The fraction of the circle is $\frac{90}{360}=\frac{1}{4}$90360=14.

More generally, If the angle of the sector is $\theta$`θ`, then the fraction of the circle is represented by

$fraction=\frac{\theta}{360}$`f``r``a``c``t``i``o``n`=`θ`360 (due to there being $360$360° in a circle).

**Question**: Find the area and circumference of a sector with central angle of $126$126° and radius of $7$7cm. Evaluate to $2$2 decimal places.

**Think**: What fraction is this sector of a whole circle? What are the rules for circumference and area?

**Do**: This sector is $\frac{126}{360}=0.35$126360=0.35 of a circle.

Circumference of a whole circle is $C=2\pi r$`C`=2π`r`, so the perimeter of the sector is

$0.35\times2\pi r$0.35×2πr |
$=$= | $0.35\times2\pi\times7$0.35×2π×7 |

$=$= | $0.35\times14\pi$0.35×14π | |

$=$= | $4.9\times\pi$4.9×π | |

$=$= | $15.39$15.39 cm (rounded to $2$2 decimal places) |

Area of a circle is $A=\pi r^2$`A`=π`r`2, so the area of the sector is

$0.35\times\pi r^2$0.35×πr2 |
$=$= | $0.35\pi\times7^2$0.35π×72 |

$=$= | $17.15\pi$17.15π | |

$=$= | $53.88$53.88 cm^{2} (rounded to 2 decimal places) |

Consider the sector below.

Calculate the perimeter. Give your answer correct to two decimal places.

Calculate the area. Give your answer correct to two decimal places.

Consider the sector below.

Calculate the perimeter. Round your answer to two decimal places.

Calculate the area. Round your answer to two decimal places.

A goat is tethered to a corner of a fenced field (shown). The rope is $9$9 m long. What area of the field can the goat graze over?

Give your answer correct to two decimal places.