NSW Mathematics Standard 11 - 2020 Edition

4.06 Area of circles and sectors

Lesson

To find the area of a circle, we need to know its radius, $r$`r`. The area, $A$`A`, can be found using the formula:

Area | $=$= | $\pi\times\text{radius }^2$π×radius 2 |

$A$A |
$=$= | $\pi r^2$πr2 |

Let's start with the area of a circle being equal to $\pi r^2$π`r`2. Remember when we found the perimeter of a sector, we found the fraction of the total circumference for the arc length. It makes sense, then, that the area of a sector is some fraction of that full area. Sometimes it is easy to recognise, for instance a semicircle is half of the total area. Or maybe you might recognise a quarter, a third, or three quarters quite easily.

*Semicircle*

$\text{Area of semicircle}=\frac{1}{2}\times\pi r^2$Area of semicircle=12×π`r`2

We can make a formula for any fraction depending on the angle, $\theta$`θ`, that subtends the arc at the centre.

Minor Sector |
Major Sector |

$A=\frac{\theta}{360^\circ}\times\pi r^2$A=θ360°×πr2 |
$A=\frac{\theta}{360^\circ}\times\pi r^2$A=θ360°×πr2 |

The fraction at the beginning of the formula, $\frac{\theta}{360^\circ}$`θ`360°, is the proportion of $360^\circ$360° that the sector's angle at the centre takes up.

Remember that a minor sector is smaller than a semi-circle, that is, when $\theta<180^\circ$`θ`<180°. A major sector is bigger than a semi-circle, that is, when $\theta>180^\circ$`θ`>180°.

Area of a circle and sector

**Area of a circle**:

$A$A |
$=$= | $\pi r^2$πr2 |

**Area of a sector**:

$A$A |
$=$= | $\frac{\theta}{360^\circ}\times\pi r^2$θ360°×πr2 |

Find the area of the circle shown, correct to one decimal place.

Consider the sector below.

Calculate the perimeter. Give your answer correct to $2$2 decimal places.

Calculate the area. Give your answer correct to $2$2 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.

If we combine our understanding of the area of circles and sectors with the area other shapes, we can add or subtract pieces to find unusually shaped areas. We can also map restricted areas that form a circle or sector using a fixed centre and a radius that is determined by some maximum distance. The questions below step through some strategies to solve these sorts of problems.

Find the area of the shaded region in the following figure, correct to 1 decimal place.

Find the area of the shaded region in the following figure, correct to 1 decimal place.

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 2 decimal places.

performs calculations in relation to two-dimensional and three-dimensional figures