GCSE Revision Notes Mathematics. Volume and Cylinders

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1 GCSE Revision Notes Mathematics Volume and Cylinders

2 irevise.com All revision notes have been produced by mockness ltd for irevise.com. Copyrighted material. All rights reserved; no part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, reprinting, or otherwise without either the prior written permission of irevise.com or a license permitting copying in the United Kingdom issued by the copyright licensing Agency.

3 Volume = Length height width Volume Area = Length Volume = Area width height For a cube, length = height = width Cube Volume = Length The area of one face of a cube is 20. Work out the volume of the cube. Area of square = length length 20cm 2 = length 2 cm = length of one side of cube Volume = Volume = Length Length Length Volume = = 20 = 20( ) = 20(2 ) = 40 cm In the trapezium, a = 6.5m, b = 8.3m and h = 3.2m The trapezium is the cross-section of a tunnel. The tunnel is 200 metres long.

4 Work out the volume of the tunnel. Area of trapezium = ( + ) Area = ( ) 3.2 = 23.7m 2 Volume = Area height Volume = 23.7(200) = 4740m The diagram shows a cuboid. The length is (5 + 1) cm. The width is (2 + 3) cm. The height is cm. The length is 7 cm longer than the width. Work out the volume of the cuboid. Length is 7cm longer than the width so = = 9 = 3 Height = 3cm Length = = 16cm Width = = 9cm Volume =

5 = 432cm The cuboid has been cut out of the wooden cube as shown. (i) Show clearly why the volume of wood remaining, in cubic centimetres, is 115. Volume of Cube = (5x)(5x)(5x) = 125 Volume of Cuboid = (x)(2x)(5x) = = 115 (ii) You are given that x = 3.5. Work out the volume of wood remaining. 115 = 115 = cm 3

6 Volume of a Cylinder = Cylinders is the radius and is the height or length of the cylinder. Curved surface area of a cylinder=. (This is the area of the rectangular part). Total Surface Area = + 2 (This is made up of its curved surface area plus its two circular ends). Questions The diagram shows two cylinders.

7 How many times bigger is the volume of the large cylinder than the small cylinder? Volume of a Cylinder = Volume of small Cylinder = = 144 cm 3 Volume of Large Cylinder = = 3600 cm 3 = 25 The large cylinder is 25 times bigger than the smaller cylinder The dimensions of two solid cylinders are shown in the diagrams below. (i) Calculate the ratio of the curved surface area of the smaller cylinder to the curved surface area of the larger cylinder. Curved Surface Area = Small Cylinder Curved Surface Area = Large Cylinder Curved Surface Area = = Ratio = : Divide both sides of the ratio by Ratio = 1:4

8 (ii) Calculate the ratio of the volume of the smaller cylinder to the volume of the larger cylinder. Volume = Small Cylinder Volume = Large Cylinder Volume = = Ratio = : Divide both sides by Ratio = 1: The cylindrical tank is one-quarter full of oil. 1 litre = 1000 cm 3 The radius of the base of the cylinder is 90 cm. The height of the cylinder is 200 cm. Work out the number of litres of oil in the tank. Volume = Volume of whole tank = = cm 3 The oil occupies one quarter of the volume of the whole tank = π cm 3 Since 1litre = 1000cm 3

9 = 405π litres Using, the number of litres of oil in the tank = litres 15.4 The diagram shows a cylinder of radius r cm and height 4r cm. (i) Work out a formula for the volume, V of the cylinder in terms of π and r. Volume = Volume = = 4 (ii) Work out the volume of the cylinder when r = 8. 4 = cm A spherical golf ball has a diameter of 4 cm. (i) Find the volume of the golf ball in terms of. radius = (diameter) radius = (4) =2cm Volume of a sphere= Volume of golf ball =

10 Volume of golf ball = cm 3 (ii) A cylindrical hole on a golf course is 10 cm in diameter and 12 cm deep. The hole is half full of water. Calculate the volume of water in the hole, in terms of radius = (diameter) radius = (10) =5cm Volume of a cylinder = Volume of cylinder = Volume of Cylinder = 300 cm 3 The hole is half full of water. Volume of water in the hole = (300 ) = 150 cm 3 (iii) The golf ball is dropped into the hole. Find the rise in the level of the water, correct to two decimal places. Volume of water + golf ball = Volume of water + golf ball = cm 3 We want to find the new height of the water Volume of a cylinder = = = cm =

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