1. Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Year 9 Arcs and Sectors
Dr J Frost
Objectives: Be able to calculate the area of a sector, the length of an arc, and solve other problems involving area and volume.
Last modified: 5th May 2015

2. Starter 4 ? 12 ? ? ? ? 4 Area of shaded region = 4 – π Area = 4π
Give your answers in terms of π . 4
Area of shaded region = 4 – π ? 12
Area = 4π ?
Perimeter = 8 + 2π ?
Area = 18π ?
Perimeter = π ? 4

3. Typical GCSE example ? 𝑃=3𝑥+ 𝜋𝑥 2 Edexcel March 2012
What is the perimeter of the shape?
𝑃=3𝑥+ 𝜋𝑥 2 ?

4. Exercise 1
Find the perimeter and area of the following shapes in terms of the given variable(s) and in terms of . (Copy the diagram first)
(Triangle is equilateral) N
Area: 33 – 
Perim: 63 + 2 ? 2 1 ? 1 2x 2x 3x 3 2x
Perimeter: 8𝑥+𝜋𝑥
Area: 6 𝑥 𝜋 𝑥 2 ? 5
Perimeter: 6𝑥+𝜋𝑥
Area: 4 𝑥 2 − 1 2 𝜋 𝑥 2 ? ? ? 4 5 2
Perimeter: 10+𝜋
Area: 6+𝜋 8
Perimeter: 2+3𝜋
Area: 4− 𝜋 2 ? ? ? ? 6 2
Perimeter: 14+5𝜋
Area: 𝜋 ? ?

5. Arcs and Sectors ! ? Area of circle: = πr2 ? Circumference of circle:θ r
Proportion of circle shaded: =
_θ_
360 ?
Sector ? ! ?
_θ_
360
_θ_
360 ?
Area of sector = πr2 ×
Length of arc = 2πr ×

6. Test Your Understanding1 N
Sector area = 10.91 ?
Area = 20
50°
Arc length = 4.36 ? 5
135° 2
(Hint: Plug values into your formula and rearrange)
Sector area = 4.04cm2 ?
105°
Radius = 4.12 ?
Arc length = 3.85cm ?
2.1cm

7. Exercise 2
Find the area and perimeter of the following sectors. Give your answers to 3sf. 1 2
Chinners Bolt runs round the corner of the semicircular part of the track with radius 120m. He goes 70 around the bend in a time of 15.3s. What was his speed?
9.58m/s
70
120m ?
80
162
3cm 3
The area of the sector is 12cm2 and its radius 5cm. What is the angle ?
55.0
6cm
Area = 6.28cm2
Perimeter = 10.2cm ?
Area = 50.9cm2
Perimeter = 29.0cm  ? ?
5cm
Rex and Max want the same amount of dog pizza. Rex takes a slice from a large pizza. Max takes a slice from a small pizza of half the radius. How many times bigger is the angle between the straight sides of Max’s slice? times bigger 4
125
32
10m 2m ?
Area = 1.12m2
Perimeter = 5.12m ?
Area = 109m2
Perimeter = 41.8m ? N
A sector of angle 270° and radius 10cm is used as the net of a cone (with no base). What is its volume? (Note that 𝑉 𝑐𝑜𝑛𝑒 = 1 3 𝜋 𝑟 2 ℎ )
Slant height of cone = 10cm Arc length = 𝟑 𝟒 × 𝟐𝝅×𝟏𝟎 =𝟏𝟓𝝅
Radius of cone = 𝟏𝟓𝝅 𝟐𝝅 = 𝟏𝟓 𝟐
Height of cone = 𝟏𝟎 𝟐 − 𝟏𝟓 𝟐 𝟐 = 𝟓 𝟕 𝟐
Volume = 𝟏 𝟑 ×𝝅× 𝟏𝟓 𝟐 𝟐 × 𝟓 𝟕 𝟐 = 𝟑𝟕𝟓 𝟕 𝝅 𝟖 𝒐𝒓 𝟑𝟖𝟗.𝟔𝒄 𝒎 𝟑 ?
87
300
3mm
7km
Area = 37.2km2
Perimeter = 24.6km
Area = 23.6mm2
Perimeter = 21.7mm ? ?