1. Year 7 Volumes Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 29 th June 2016 Objectives: (a) Know the names of common solids and how they are related. (b) Understand what is meant by volume and surface area. (c) Find the volume and surface area of a cuboid. (d) Find the volume of prisms (including cylinders).

2. Categorising 3D shapes  3D Shapes Polyhedron Non-Polyhedron = “is a type of” A 3D shape with flat faces and straight edges. ? Definition Regular Polyhedron All faces the same regular polygon. ? Definition Platonic Solids Kepler-Poinsot Polyhedra Cube Cuboid Cylinder Sphere PyramidPrism Cone Concave (i.e. inwards-ey) Convex (i.e. outwards-ey) 6 rectangular faces ? Definition Solid with two congruent parallel sides (the cross-section) ? Definition * All points on base connected to ‘apex’ (top) Circular cross- section Circular base All points equidistant from centre, in 3- dimensions. ? Definition ? ? ? ? 6 square faces ? Definition (Don’t write) * One definition of a prism requires the cross-section to be a polygon (but circles are not polygons)

3. Platonic Solids A Platonic Solid is a (convex) polyhedron where all faces are the same regular polygon. FacesEach face Name Edges VerticesDiagram Cube Tetrahedron Octahedron Dodecahedron Icosahedron Square Triangle Pentagon Triangle 6 4 8 12 20 12 6 30 8 4 6 20 12 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

4. Euler’s Polyhedral Formula Faces Edges Vertices 6 4 8 12 20 12 6 30 8 4 6 20 12 ?

5. Edges, Faces and Vertices Example: How many diagonals does an octagon have? ?

6. [IMC 2015 Q7] A tetrahedron is a solid figure which has four faces, all of which are triangles. What is the product of the number of edges and the number of vertices of the tetrahedron? Solution: 24 [IMC 2006 Q5] A solid ‘star’ shape is created by gluing a square-based pyramid, in which each edge is of length 1 unit, precisely onto each face of a cube of edge 1 unit. How many faces does this ‘star’ have? Solution: 24 [JMC 2004 Q8] A solid square-based pyramid has all of its corners cut off, as shown. How many edges does the resulting shape have? Solution: 24 Exercise 1 1 2 3 4 5 6 ? ? ? ? ? ?

7. [IMC 1999 Q14] Which of the following statements is false? A an octagon has twenty diagonals B a hexagon has nine diagonals C a hexagon has four more diagonals than a pentagon D a pentagon has the same number of diagonals as it has sides E a quadrilateral has twice as many diagonals as it has sides Solution: E [IMC 1997 Q24] A regular dodecahedron is a polyhedron with twelve faces, each of which is a regular pentagon. A space diagonal of the dodecahedron is a line segment which joins two vertices of the dodecahedron which do not lie in the same face. How many space diagonals are there in the dodecahedron? Solution: 100 7  ? ?

8. Volume and Surface Area of a Cuboid Volume is the amount of space an object takes up. Surface Area is the total area across the surface. (you could think of as the amount of wrapping paper required) ? ? Volume of cuboid = width x height x length ? General Formula

9. Test Your Understanding 6cm 3cm 2cm 50cm 10cm ? 1 2 ?

10. Breaking up Solids 10cm The Box of Happiness 3cm 2cm 4cm To find the volume, what might be our strategy here? A B C A B C ? ? ? ?

11. Test Your Understanding 7cm 5cm 6cm 4cm 1cm ? ?

12. Exercise 2 Find the volume and surface area of the following cuboid. 3cm 1cm 2cm Find the volume and surface area of the following cuboid. 5m 4m 7m 2 3 1 4 Find the volume of the following solid. 16cm 2 1 2 3 5 4 ? ? ? ? ? ? ? 1

13. Exercise 2 3m Find the volume of the following solid. 290cm 3 6 2m 8m 5m [JMO 2006 A2] The perimeter of this net of a cube is 42cm. What is the volume of the cube? Solution: 27 cm 3 7 8 [JMC 2007 Q17] Just William’s cousin, Sweet William, has a rectangular block of fudge measuring 2 inches by 3 inches by 6 inches. He wants to cut the block up into cubes whose side lengths are whole numbers of inches. What is the smallest number of cubes he can obtain? Solution: 15 [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 small cubes are in the remaining solid? Solution: 46 9 10 ? ? ? ? ?

14. Exercise 2 11 [JMC 2011 Q20] One cube has each of its faces covered by one face of an identical cube, making a solid as shown. The volume of the solid is 875cm 3. What, in cm 2, is the surface area of the solid? Solution: 750 [JMO 2005 A10] A closed rectangular box is a double ‘cube’, in which the top and bottom are squares, and the 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? Solution: 135 cm 2 12 13 14 15 ? ? ? ? ?

15. Exercise 2 15 ?

16. Prisms A prism is a solid where you see the same ‘cross section’ anywhere you slice it. You can think of a prism as the tube formed when Playdough is forced through a shape. The cross section will be the shape it is forced through! (star, square, etc) length = 3 How many cubes is each cross-section (i.e. layer) made up of? 8 How many cubes are there in the solid in total? 24 Can you think of a suitable formula in general? ? ? ?

17. Examples 4cm 3cm 7cm 4cm 8cm 3cm 5cm ? ? ? ?

18. Check Your Understanding 10m 5m 4m 6m ? ? 13m 10m 4m ? A harder one if you finish…

19. Cylinders A cylinder is just a prism with a circular cross-section. So the maths is exactly the same! ? ?

20. Example 6cm 10cm ?

21. Check Your Understanding TOOFPASTE TM 3cm 14cm 1  40cm 50cm 10cm 15cm 1.2m ? ?

22. Example 3 3m 7m 2 A prism has a cross- sectional area of 7m 2 and a length of 3m. What is its volume? Solution: 21m 3 1 2 Find the volume of this prism. 4cm 5cm 10cm Solution: 100 cm 3 5cm 7cm 11cm 4cm 20cm Find the volume of this prism. Solution: 1180cm 3 3 Find the volume of this cylinder (to 3dp). Solution: 125.66 cm 3 RightUpYourAlley TM are manufacturing a new cylindrical toilet roll with the pictured dimensions. As usual the centre is hollow. Find the volume of paper. 907.13 cm 3 4 6 5 ? ? ? ? ? Find the volume of this prism. Solution: 40m 3 ?

23. Example 3 1.3m 1m 2m 0.5m 22 11 7 A square-based prism is 320cm 3 in volume. If its length is 20cm, what is the side of the square? Solution: 4cm [IMC 2010 Q17] Last year Gill’s cylindrical 21 st 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 year’s cake? A 12:1B 7:1C 6:1D 4:1E 3:1 Solution: A 8 ? ? ? ? ?