Arcs & Sectors (Edexcel IGCSE Further Pure Maths): Revision Note

• Be careful on arc length questions

  • Finding the arc length may only be part of a larger question

  • For example finding the total perimeter of a circle sector

• Make sure you are using the formula that matches the angle measure (degrees or radians)

A circular pizza has had a slice cut from it. The slice is in the shape of a sector, with the angle at the centre being 38°. The radius of the pizza is 12 cm.

(a) Find the length of the outside crust of the slice of pizza (the minor arc), giving your answer correct to 3 significant figures.

Drawing a diagram can help with questions like this

Use the arc length formula (in the degrees form) with $r=12$ and $\theta =38$
(You could also convert $38°$ to radians and use the radians version of the formula)

$l=\frac{38}{360}\times 2\pi \left(12\right)=\frac{38\pi }{15}=7.958701...$

Round to 3 significant figures

(Note that $\frac{38\pi }{15}$ cm is the exact value answer)

$\mathbf{7}\mathbf{.}\mathbf{96}\mathbf{cm}\left(3s.f.\right)$ 

(b) Find the perimeter of the remaining pizza, giving your answer correct to 3 significant figures.

Drawing a diagram can help

The remaining pizza will be in the shape of a major sector

The angle at the centre will be  $360-38=322°$

The perimeter will include both the major arc and the two radii

Use the arc length formula (in the degrees form) with $r=12$ and $\theta =322$

$\mathrm{perimeter}=\left(\frac{322}{360}\times 2\pi \left(12\right)\right)+12+12=\frac{322\pi }{15}+24=91.439522...$

Round to 3 significant figures

(Note that $\frac{322\pi }{15}+24$ cm is the exact value answer)

91.4 cm (3 s.f.)

A sector of a circle has a radius of 7cm and an angle at the centre of $\frac{\mathrm{\pi }}{6}$ radians. Find the perimeter of the sector, giving your answer as an exact value.

Drawing a diagram can help

The perimeter of sector will include both the length of the arc and the two radii

Use the arc length formula (in the radians form) with $r=7$ and $\theta =\frac{\pi }{6}$

$\mathrm{perimeter}=\left(7\right)\left(\frac{\pi }{6}\right)+7+7=\frac{7\pi }{6}+14$

The question asks for an exact value answer, so just write that down (with units!)

$\frac{\mathbf{7}\mathit{\pi }}{\mathbf{6}}\mathbf{+}\mathbf{14}\mathbf{cm}$

• Make sure you are using the formula that matches the angle measure (degrees or radians)

Jamie has divided a circle of radius 50 cm into two sectors: a minor sector of angle 100°, and a major sector of angle 260°. He is going to paint the minor sector blue and the major sector yellow.

(a) Find the area Jamie will paint blue, giving your answer correct to 3 significant figures

Drawing a diagram can be helpful in questions like this

Use the sector area formula (degrees form) with $r=50$ and $\theta =100$

(You could also convert $100°$ to radians and use the radians version of the formula)

$A=\frac{100}{360}\times \pi {\left(50\right)}^2=\frac{6250\pi }{9}=2181.6615...$

Round to 3 significant figures

(Note that $\frac{6250\pi }{9}$ cm² is the exact value answer) 

$\mathbf{2180}{\mathbf{cm}}^{\mathbf{2}}\left(3s.f.\right)$

(b) Find the area Jamie will paint yellow, giving your answer correct to 3 significant figures.

This will be the rest of the circle from the preceding diagram

Use the sector area formula (degrees form) with $r=50$ and $\theta =260$

$A=\frac{260}{160}\times \pi {\left(50\right)}^2=\frac{16250\pi }{9}=5672.3200...$

Round to 3 significant figures

(Note that $\frac{16250\pi }{9}$ cm² is the exact value answer)

$\mathbf{5670}{\mathbf{cm}}^{\mathbf{2}}\left(3s.f.\right)$

A sector of a circle has a radius of 7 cm and an angle at the centre of $\frac{\mathrm{\pi }}{6}$ radians. Find the area of the sector, giving your answer as an exact value.

Drawing a diagram can help

Use the sector area formula (radians form) with $r=7$ and $\theta =\frac{\pi }{6}$

$A=\frac{1}{2}{\left(7\right)}^2\left(\frac{\pi }{6}\right)=\frac{49\pi }{12}$

The question asks for an exact value answer, so just write that down (with units!)

$\frac{\mathbf{49}\mathbf{\pi }}{\mathbf{12}}{\mathbf{cm}}^{\mathbf{2}}$