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Year 7 Volumes Year 7 Volumes Exercise 1 – Vertices, Edges and Faces 1. [IMC 2015 Q7] A 6. [IMC 2015 Q17] The football tetrahedron is a solid shown is made by sewing figure which has four together 12 black pentagonal faces, all of which are panels and 20 white triangles. What is the product of the hexagonal panels. There is a join number of edges and the number of wherever two panels meet along an vertices of the tetrahedron? edge. How many joins are there? 2. [IMC 2006 Q5] A solid ‘star’ shape is created by gluing a square-based 7. [IMC 1999 Q14] Which of the pyramid, in which each edge is of following statements is false? length 1 unit, precisely onto each face A an octagon has twenty diagonals of a cube of edge 1 unit. How many B a hexagon has nine diagonals faces does this ‘star’ have? C a hexagon has four more diagonals than a pentagon 3. [JMC 2004 Q8] A solid D a pentagon has the same number square-based pyramid of diagonals as it has sides has all of its corners cut E a quadrilateral has twice as many off, as shown. How diagonals as it has sides many edges does the resulting shape have? [IMC 1997 Q24] A regular dodecahedron is a polyhedron with 4. [JMO 2011 A5] The base of a pyramid twelve faces, each of which is a has 푛 edges. In terms of 푛, what is the regular pentagon. A difference between the number of space diagonal of the edges of the pyramid and the number dodecahedron is a line of its faces? segment which joins two vertices of the 5. [JMC 2002 Q19] The number of dodecahedron which do not lie in the diagonals of a regular polygon equals same face. How many space diagonals twice the number of sides. How many are there in the dodecahedron? sides has the polygon? Exercise 2 1. Find the volume and surface area of 2. Find the volume and surface area of the following cuboid. the following cuboid. 3. The area of one side of a cube is 16m2. What is its volume? www.drfrostmaths.com 4. Find the volume of the following solid. 10. [JMO 1999 A6] A cube is made of 64 small cubes. Three holes are made, with each hole perpendicular to two faces and passing right through the cube. The shape and position of each hole is shown in the diagram. How many 5. A cuboid milk carton has a square small cubes are in the remaining base of 8cm by 8cm and is 20cm solid? 3 in height. I pour 400cm of milk 11. [JMC 2008 Q14] A solid wooden cube out into a glass. What is the is painted blue on the outside. The height of milk left in the carton? cube is then cut into eight smaller 6. Find the volume of the following solid. cubes of equal size. What fraction of the total surface area of these new cubes is blue? 12. [JMC 2002 Q20] Sally has 72 small wooden cubes, each measuring 1푐푚 × 1푐푚 × 1푐푚. She arranges them all so that they form a cuboid. Given that the perimeter of the base 7. [JMO 2006 A2] The of the cuboid is 16cm, what is its perimeter of this net of a height? cube is 42cm. What is the 13. [IMC 2011 Q8] A square piece of card volume of the cube? has a square of side 2cm cut out from each of its corners. The remaining 8. [JMO 2013 A3] The card is then folded along the dotted solid shown is made by lines shown to form an open box gluing together four whose total internal surface area is 2 1cm × 1cm × 1cm 180 cm . What is the volume of the 3 cubes. open box in cm ? What is the total surface area of the 14. [JMC 2011 Q20] One cube solid? has each of its faces covered by one face of an identical 9. [JMC 2007 Q17] Just William’s cousin, cube, making a solid as Sweet William, has a rectangular shown. The volume of the solid is 3 2 block of fudge measuring 2 inches by 875cm . What, in cm , is the surface 3 inches by 6 inches. He wants to cut area of the solid? the block up into cubes whose side 15. [JMO 2005 A10] A closed rectangular lengths are whole numbers of inches. box is a double ‘cube’, in which the What is the smallest number of cubes top and bottom are squares, and the he can obtain? height is twice the width. The greatest distance between any two points of this box is 9 cm. What is the total surface area of the box? 16. IMC 1998 Q20] The total length of all the edges of a cube is 퐿 cm. If the surface area of the cube has the same 퐿 2 numerical value cm , what is its 3 volume in cm ? A 1 B 퐿 C 2 D 퐿3 E 8 www.drfrostmaths.com Exercise 3 7. A square-based prism is 320cm3 in volume. If its length is 20cm, what is 1. A prism has a cross-sectional area of the side of the square? 7m2 and a length of 3m. What is its 8. [IMC 2010 Q17] Last year Gill’s volume? cylindrical 21st birthday cake wasn’t big enough to feed all her friends. This year she will double the radius and triple the height. What will be the ratio of the volume of this year’s birthday cake to the volume of last 2. Find the volume of this prism. year’s cake? A 12:1 B 7:1 C 6:1 D 4:1 E 3:1 Find the volumes of the following shapes, giving your answers in terms of 푥 (and 휋 where applicable) 3. Find the volume of this prism. [Edexcel] The pond is completely full of water. Sumeet wants to empty the pond so he can clean it. Sumeet uses a pump to empty the pond. The 4. Find the volume of this cylinder (to volume of water in the pond 3dp). decreases at a constant rate. The level of the water in the pond goes down by 20cm in the first 30 minutes. Work out how much more time Sumeet has to wait for the pump to empty the pond completely. 5. Find the volume of this prism. 6. RightUpYourAlleyTM are manufacturing a new cylindrical toilet roll with the pictured dimensions. As usual the centre is hollow. Find the volume of paper. www.drfrostmaths.com .
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