Visualising Solid Shapes

We live in a three-dimensional world, but we often have to show solid objects on flat paper. In this chapter you will learn that a solid looks different from different positions, and you will draw its front view, side view and top view, as for a cylinder, a cone or a pyramid. You will compare maps with pictures, learning that a map shows places from above, uses symbols and is drawn to a scale, so that 1 cm on the map may represent 500 m on the ground. You will then study polyhedrons, solids whose faces are polygons, such as cubes, prisms and pyramids, and distinguish convex polyhedrons from others and regular polyhedrons from irregular ones. You will count the faces, vertices and edges of polyhedrons and verify Euler's formula F + V - E = 2. You will also draw nets that fold into solids and sketch solids on isometric dot paper.

Detailed Worksheet: Visualising Solid Shapes Section A - Definitions (10 marks) 1. What is a polyhedron? Is a cylinder a polyhedron? Give a reason. (2 marks) 2. Define a convex polyhedron and a regular polyhedron, giving one example of each. (2 marks) 3. Distinguish between a prism and a pyramid, giving one example of each. (2 marks) 4. State Euler's formula for polyhedrons, explaining what F, V and E stand for. (2 marks) 5. What is the net of a solid? Name the solid whose net consists of one square and four congruent triangles. (2 marks) Section B - Calculations and Applications (15 marks) 6. Verify Euler's formula for (i) a triangular prism (ii) a square pyramid (iii) a cube. (3 marks) 7. Using Euler's formula, find: (i) the number of edges of a polyhedron with 8 faces and 6 vertices (ii) the number of vertices of a polyhedron with 20 faces and 30 edges. (iii) Can a polyhedron have 10 faces, 20 edges and 15 vertices? Give a reason. (3 marks) 8. Find the number of faces, vertices and edges of (i) a hexagonal prism (ii) a hexagonal pyramid, and verify Euler's formula for each. (3 marks) 9. A town map uses the scale 1 cm represents 500 m. (i) The school is 4.5 cm from Rani's house on the map; find the actual distance. (ii) The park is 1.75 km from her house; find the distance on the map. (3 marks) 10. A solid is built from unit cubes: a bottom layer of 3 by 3 cubes, a middle layer of 2 by 2 cubes and a single cube on top. How many cubes are used? How many unit squares are seen in the top view? (3 marks) Section C - Diagrams (10 marks) 11. Draw the front view, side view and top view of (i) a cylinder standing on its base (ii) a cone standing on its base (iii) a square pyramid standing on its base. (4 marks) 12. Draw one net of a cube and one net of a triangular prism. Label the faces that will be opposite each other in the cube. (3 marks) 13. On isometric dot paper, sketch a cuboid of length 4 units, breadth 3 units and height 2 units. Also draw its top view. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Six squares joined in a straight line cannot be folded into a cube. Explain why. Draw two different arrangements of six squares that do fold into a cube, and state how many different nets a cube has in all. (5 marks) 15. There are only five regular polyhedrons. Name them and write the number of faces, vertices and edges of each. Verify Euler's formula for any two of them, and explain why a cube is a regular polyhedron but a cuboid with unequal edges is not. (5 marks) 16. Write two differences between a map and a picture. Raju walks 300 m north from his house, then turns right and walks 400 m to his school. Draw a map of his route using the scale 1 cm represents 100 m, and find the straight-line distance from his house to the school. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw all views, nets and maps neatly with a ruler and pencil, and state the scale used on every map.