Measures and solids · Ages 12–13
3-D figures worksheet
Covers: Prisms, pyramids and round solids, Cross-sections, Nets of solids, Drawing solids on paper, Three views, Euler's formula and regular polyhedra
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3-D figures worksheet
Measures and solids · Ages 12–13
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.A pyramid on a 6-sided base has faces, edges and vertices.
Fill in each blank
Answer:
2.The solid is built of equal cubes, every column standing on the ground. How many cubes are there?
Answer:
3.An oblique drawing of a cube with 11 cm edges: the front drawn its true size, the 4 edges going back half their real length. What is the total length of the 12 edges in the drawing (solid and dashed), in cm?
Answer:
4.The solid is built of cubes, every column standing on the ground. How many squares do you see from above?
Answer:
5.A polyhedron has 6 faces and 10 edges. By Euler's formula, its vertices V = 2 − + = .
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
6.Which solid has 5 faces, 8 edges and 5 vertices?
- A
- B
- C
- D
Choose the right picture
Answer:
7.A plane cuts the solid in the picture; the shaded part is the cross-section. What shape is it?
- A
- B
- C
- D
Choose the right picture
Answer:
8.Wherever a solid is cut parallel to its base, the cross-section is the same 5-sided polygon. What is the solid?
- A. A prism on a 5-sided base
- B. A cylinder
- C. A prism on a 7-sided base
- D. A pyramid on a 5-sided base
Choose the right answer
Answer:
9.The six squares in the picture fold into a cube.
True or false?
Answer:
10.Folded into a cube, the faces marked 3 and 4 are opposite each other.
True or false?
Answer:
11.The solid is built of cubes, every column standing on the ground. Which shape do you see from the front?
- A
- B
- C
- D
Choose the right picture
Answer:
12.A polyhedron has 9 faces, 9 vertices and 16 edges, and these numbers fit Euler's formula.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.A sharpened pencil is a prism on a 3-sided base with a pyramid on one end. How many faces does it have?
Working:
Answer:
14.A block of tofu 12 cm long, 8 cm wide and 9 cm high lies flat. It is cut 2 times parallel to its base. By how many cm² has the total surface area grown?
Working:
Answer:
15.From each corner of a 22 cm square of card a 2 cm square is cut, and the rest folded into an open box. What is its volume, in cm³?
Working:
Answer:
16.On grid paper, an oblique drawing of a cuboid 8 squares long, 10 wide and 4 high, its front the length × height face, edges going back drawn half length. What do its 12 edges (dashed ones too) add up to, in squares?
Working:
Answer:
17.A solid built of cubes: the picture shows its front view (top) and its view from above (bottom, the back at the top). It has at least cubes and at most .
Fill in each blank
Working:
Answer:
18.A gem has 18 triangular faces, two pyramids stuck base to base. How many vertices does it have?
Working:
Answer:
Show the answers and working
3-D figures worksheet: Answers and working
Part A: Working out and filling in
1. 7, 12, 7
Faces: the base and 6 triangles, 7; edges: 6 round the base and 6 slanting, 12; vertices: 6 round the base and 1 at the top, 7.
2. 8 cubes
Count each column's height, hidden cubes too: front 1 + 1 + 0, back 2 + 2 + 2, 8 in all.
3. 110 cm
The 8 edges of the front and back faces are drawn true: 8 × 11 = 88; the 4 going back at half: 4 × 5.5 = 22; 88 + 22 = 110 cm in all.
4. 8 squares
From above, each place with cubes shows one square however tall: 8 in all.
5. 6, 10, 6
F + V − E = 2 rearranges to V = 2 − F + E = 2 − 6 + 10 = 6.
Part B: Choose, or true or false
6. D
Tell prism from pyramid, then count the base's sides n: a prism has n + 2 faces, 3n edges, 2n vertices; a pyramid n + 1 faces, 2n edges, n + 1 vertices.
7. D
A prism or cylinder cut parallel to its base gives the base again; a pyramid or cone gives the base's shape, smaller near the top; a sphere always gives a circle; a lying triangular prism cut level gives a rectangle.
8. A. A prism on a 5-sided base
The same cross-section all the way along marks a prism; 5 sides makes it a prism on a 5-sided base.
9. False
A cube's net never has a 2 × 2 block or more than 4 squares in a line; folded, the six squares take a face each, none overlapping.
10. True
Opposite faces never touch, and squares one apart in a line are opposite. Folded, the face opposite 3 is 4.
11. C
From the front, each column across shows only its tallest stack: 4, 3 and 1 high, left to right.
12. True
Put them in F + V − E: 9 + 9 − 16 must equal 2. The edges that fit are 9 + 9 − 2 = 16.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 7 faces
The prism's 3 sides and its end base, and the pyramid's 3 triangles; where they join is no face: 3 + 1 + 3 = 7.
14. 384 cm²
Each cut makes two new faces, each the base again, 12 × 8 = 96: 2 × 2 × 96 = 384.
15. 648 cm³
The base's side is 22 − 2 × 2 = 18 cm (2 off each end), the height 2 cm: 18 × 18 × 2 = 648.
16. 68 squares
The front and back faces' lengths and heights are true: 4 × 8 + 4 × 4; the 4 edges going back are 5 each: 4 × 5; 68 in all.
17. 6, 7
Each column across: the top view gives its places, the front view its tallest. Fewest: one place that tall and the rest 1 each; most: every place that tall. 6 at least, 7 at most.
18. 11 vertices
Each pyramid has 18 ÷ 2 = 9 triangles on a 9-sided base: its 9 base corners and two tips make 11. Or by Euler: edges 18 × 3 ÷ 2 = 27, V = 2 − 18 + 27 = 11.
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