Arcs & Sectors (SQA National 5 Maths): Revision Note

From your National 4 Maths course, you should be familiar with the formula for the circumference, $C$, of a circle with radius $r$:

$C=2\pi r$

Make sure you know this, as it is not given to you in the Formulae List on the exam paper.

The diagram below shows a sector of a circle, centre $\mathrm{C}$.

The radius of the circle is 24 centimetres.

Calculate the length of the major arc $\mathrm{AB}$.

Give your answer correct to one decimal place.

Answer:

Divide the angle by 360 to form a fraction

• Here $\theta =210°$

$\frac{\theta }{360}=\frac{210}{360}$

Use $C=2\pi r$ to find the circumference of the full circle

• Here $r=24$

$2\pi r=2\pi \times 24$

Multiply the fraction by the circumference

• This is the same as substituting $\theta =210$ and $r=24$ into the formula $\mathrm{arc}\mathrm{length}=\frac{\theta }{360}\times 2\pi r$

• If you remember that formula you can skip right to this step!

 $\mathrm{arc}\mathrm{length}=\frac{210}{360}\times 2\pi \times 24$

Use your calculator to work out that value 

$=87.964594...$

Round your answer to 1 decimal place, as required

88.0 cm (1 d.p.)

From your National 4 Maths course, you should be familiar with the formula for the area, $A$, of a circle with radius $r$:

$A=\pi r^2$

Make sure you know this, as it is not given to you in the Formulae List on the exam paper.

The diagram below shows a sector of a circle, centre $\mathrm{C}$.

The radius of the circle is 21 centimetres and angle $\mathrm{ACB}$ is 150°.

Calculate the area of the sector.

Give your answer correct to one decimal place.

Answer:

Divide the angle by 360 to form a fraction

• Here $\theta =150°$

$\frac{\theta }{360}=\frac{150}{360}$

Use $A=\pi r^2$ to find the area of the full circle

• Here $r=21$

$\pi r^2=\pi \times {21}^2$

Multiply the fraction by the area

• This is the same as substituting $\theta =150$ and $r=21$ into the formula $\mathrm{sector}\mathrm{area}=\frac{\theta }{360}\times \pi r^2$

• If you remember that formula you can skip right to this step!

 $\mathrm{sector}\mathrm{area}=\frac{150}{360}\times \pi \times {21}^2$

Use your calculator to work out that value 

$=577.267650...$

Round your answer to 1 decimal place, as required

577.3 cm² (1 d.p.)