Area of Sector and Arc Length
Introduction
A sector is a part of a circle bounded by two radii and one of the arcs formed by the intersections of these is with the circumference.
The angle subtended at the centre of the circle by the arc of the sector is the angle of the sector.
When a circle is divided into two sectors the larger sector is the major sector and the small one is the minor sector.
The two formulas that are needed to calculate the area of sector and arc length are as follows:
Arc length = \frac{\theta}{360} \times 2 \pi r
Area of sector = \frac{\theta}{360} \times \pi r^2
These are two formulas that you will need to remember as these are not on the formula sheet for GCSE Maths. This applies to whether doing the foundation paper or higher paper.
Below are some examples on area of sector and arc length and also some questions for you to attempt on your own. Do make sure that you are understanding the methods that are being used and keep trying the questions repeatedly until you fully understand the process.
Area of Sector and Arc Length - Example 1
Take a look at the following question:
Area of sector is given by:
\frac{150}{360} \times \pi \times 13^2=221.108 \ldots=221 \mathrm{~cm}^2This was quite a straightforward question because all that was required was a simple substitution. Make sure you are giving answers as the question wants.
Area of Sector and Arc Length - Example 2
Take a look at the following question:
Here you first of all need to find the angle of the sector:
\begin{aligned} & 15=\frac{\theta}{360} \times 2 \times \pi \times 10 \\ & \frac{15 \times 360}{20 \pi}=\theta \rightarrow \theta=85.94^{\circ} \end{aligned}Now the area of the sector can be found:
\text { Area }=\frac{85.94}{360} \times \pi \times 10^2=74.958 \ldots=75 \mathrm{~cm}^2
Area of Sector and Arc Length – Example 3
Take a look at the following question:
Here arc length AB is given by:
\frac{60}{360} \times 2 \times \pi \times 12=4 \pi \mathrm{cm}Area of Sector and Arc Length – Question Practice 1
Try the following questions on your own before looking at the solution:
Area of Sector and Arc Length – Question Practice Solution
First you need to calculate the arc length and remember to leave your answer in terms of \pi
The arc length is given by:
\frac{120}{360} \times 2 \times \pi \times 6=4 \pi \mathrm{cm}Now the perimeter is distance all the way around an object so the perimeter of the arc is:
6+6+4 \pi=12+4 \pi \mathrm{cm}Area of Sector and Arc Length – Question Practice 2
Try the following questions on your own before looking at the solution:
In this question you need to calculate the area of the sector and also the area of the triangle.
Area of sector is found by:
\frac{40}{360} \times \pi \times 8^2=22.32888 \ldotsArea of triangle is found by:
\frac{1}{2} \times 8 \times 8 \times \sin 40=20.56920 \ldotsSo the area of the shaded part = area of sector – area of triangle is found by:
22.32888-20.56920=1.75968=1.76 \mathrm{~cm}^2As long as you are able to apply the formulas that are associated with this topic then this topic is quite straightforward. Make sure that you are show all your work when doing calculations even if such questions appear on the GCSE Maths calculator paper.
Use the diagram that is given to you and if one is not provided then you should draw a quick sketch to help you understand the question better.