Year 9 Arcs and Sectors Dr J Frost

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Presentation transcript:

Dr J Frost (jfrost@tiffin.kingston.sch.uk) Year 9 Arcs and Sectors Dr J Frost (jfrost@tiffin.kingston.sch.uk) 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

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 = 12 + 6π ? 4

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

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 2 + 1 2 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: 24+ 25 2 𝜋 ? ?

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 ×

Test Your Understanding 1 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

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? 4 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 ? ?