Year 7 Volumes Dr J Frost jfrosttiffin kingston

Year 7 Volumes Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com 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). Last modified: 29 th June 2016

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

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

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

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

Exercise 1 1 [IMC 2015 Q 7] 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 4 ? 2 [IMC 2006 Q 5] 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 ? 3 ? 5 ? 6 [JMC 2004 Q 8] 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 7 [IMC 1999 Q 14] 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 ? N [IMC 1997 Q 24] 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

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) ? ? General Formula Volume of cuboid = width x height x length ?

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

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

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

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

Exercise 2 6 Find the volume of the following solid. 290 cm 3 3 m 8 m ? 2 m 5 m 8 m 7 [JMO 2006 A 2] The perimeter of this net of a cube is 42 cm. What is the volume of the cube? Solution: 27 ? cm 3 8 ? 9 [JMC 2007 Q 17] 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 ? 10 [JMO 1999 A 6] 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 ?

Exercise 2 11 14 ? 12 [JMC 2011 Q 20] 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 875 cm 3. What, in cm 2, is the surface area of the solid? Solution: 750 ? 13 ? 15 ? [JMO 2005 A 10] 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

Exercise 2 15 ?

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) 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? length = 3 ?

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

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

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

Example 6 cm ? 10 cm

Check Your Understanding 1 3 cm TOOFPASTETM ? 14 cm 40 cm N 1. 2 m 50 cm ? 15 cm 10 cm

Example 3 1 7 m 2 3 m 2 A prism has a crosssectional area of 7 m 2 and a length of 3 m. What is its volume? Solution: 21 m 3 4 ? ? Find the volume of this prism. 5 4 cm 3 5 cm 10 cm 7 cm 4 cm 20 cm 11 cm Find the volume of this prism. Solution: ? 1180 cm 3 Find the volume of this prism. Solution: ? 40 m 3 Solution: ? 100 cm 3 5 cm Find the volume of this cylinder (to 3 dp). Solution: 125. 66 cm 3 6 Right. Up. Your. Alley. 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

Example 3 A square-based prism is 320 cm 3 in volume. If its length is 20 cm, what is the side of the square? Solution: 4 cm 7 N 2 1 m 2 m 0. 5 m ? st 8 [IMC 2010 Q 17] Last year Gill’s cylindrical 21 birthday 1. 3 m 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: 1 B 7: 1 C 6: 1 D 4: 1 E 3: 1 Solution: A ? N 1 ? ? ?