Solid shapes worksheet

Part A: Working out and filling in

1. 1. A cube has 8 vertices

   The top and the bottom have 4 vertices each: 4 × 2 = 8, so 8 vertices

2. 2. This prism has 7 faces

   Its base is a pentagon, with 5 sides. A prism has 2 bases and 5 faces round the side: 5 + 2 = 7, so 7 faces

3. 3. A pentagonal pyramid has 10 edges

   The base has 5 edges, and one runs from the top to each corner of it: 5 × 2 = 10, so 10 edges

4. 4. 153 cm²

   Surface area = the square base + four triangles. Work out the base, 9 × 9, then one triangle's area.

   Base: 9 × 9 = 81; four triangles: 4 × 9 × 4 ÷ 2 = 72; in all 81 + 72 = 153.

5. 5. 144 cm²

   A triangular prism has two triangles and three rectangles. Start with one right-angled triangle: its perpendicular sides are 3 and 4.

   All three rectangles are 11 cm long; their widths are the triangle's three sides.

   Two triangles: 2 × 3 × 4 ÷ 2 = 12; three rectangles: (3 + 4 + 5) × 11 = 132; in all 144.

Part B: Choose, or true or false

1. 6. D

   Look at the solid's base first. Cut parallel to it, how does the cross-section compare with the base?

   Cut parallel to the base, the cross-section has the base's shape.

2. 7. True

   First decide: is the solid the same width all the way up, like a prism or cylinder, or does it narrow to a point, like a pyramid or cone?

   Prisms and cylinders (cubes and cuboids too) have every slice the size of the base; pyramids and cones get smaller towards the top.

3. 8. D

   Take each solid in turn: is there any way to cut it that doesn't give a circle?

   Every point of a sphere is the same distance from its centre, so every cross-section is a circle.

4. 9. D

   A cylinder has two circular ends and a curved side that opens out into a rectangle. Check where each shape puts its two circles.

   A cylinder's net is a rectangle with a circle on each long side.

5. 10. C. 9 cm × 8 cm

   The base is length × width. Which face is opposite the base, and how do opposite faces of a cuboid compare?

   A cuboid's opposite faces match: the top is also 9 cm × 8 cm.

6. 11. B

   From above, height doesn't matter: only which places have cubes. Check each place in the front and back rows.

   From above, every place with cubes shows one square: 3 in the back row, 2 in the front.

7. 12. C

   Seen from the right, the front row is on the left and the back row on the right. Find how tall each row's tallest stack is first.

   From the right, the front row is on the left and the back row on the right: the front row is 2 high at most, the back row 4.

Part C: Word problems

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

1. 13. Tsz-ching makes a prism frame from straws. The prism's base has 5 sides, and each edge is one straw. How many straws does she need? (Answer: 15 straws)

   5 edges round the top, 5 round the bottom, 5 up the sides: 5 × 3 = 15

2. 14. Ming makes a prism frame from straws. The prism's base has 6 sides, and each edge is one straw. How many straws does he need? (Answer: 18 straws)

   6 edges round the top, 6 round the bottom, 6 up the sides: 6 × 3 = 18

3. 15. 8 cm

   The sausage is a cylinder. Cut parallel to its ends, is each slice's circle any different in size from an end?

   A cylinder's cross-sections parallel to its base match the base: 8 cm, however many slices.

4. 16. 324 cm

   The wire needed is all the edges added up. How many edges are there of each length?

   The frame is the 12 edges, 4 of each length: 4 × (24 + 28 + 29) = 324.

5. 17. 60 m²

   Only the four walls count. Two walls are 6 × 3; what are the other two? Or open all four walls out into one long rectangle.

   Only the four walls: opened out they make one rectangle, as long as the way round the room, 2 × (6 + 4) = 20, and 3 high: 20 × 3 = 60.

6. 18. 6 squares

   The painted squares are the ones seen from above. A stack shows one square however tall: count the places with blocks.

   Each place with blocks shows one square from above: 3 at the back, 3 at the front, 6 in all.