Measures and solids · Ages 14–15

Similar figures and solids worksheet

Part of: Cones & spheres

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Measures and solids · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Two similar cones are 1 cm and 6 cm high. The small cone's volume is 2 cm³. What is the big cone's volume, in cm³?
Answer:
2.Two similar solids have matching lengths in the ratio 5 : 6. Their surface areas are in the ratio : , and their volumes : .
Fill in each blank
Answer:
3.Two similar cones are 1 cm and 3 cm high. The small cone's volume is 2 cm³. What is the big cone's volume, in cm³?
Answer:
4.Two similar cylinders have volumes in the ratio 27 : 64. The small one is 6 cm high. How high is the big one, in cm?
Answer:
5.Two similar solids have surface areas in the ratio 4 : 25. The small one's volume is 48 cm³. What is the big one's, in cm³?
Answer:
6.Two similar cylinders have volumes in the ratio 1 : 8. The small one is 3 cm high. How high is the big one, in cm?
Answer:
7.Two similar solids have surface areas in the ratio 4 : 9. The small one's volume is 48 cm³. What is the big one's, in cm³?
Answer:
8.A model of a building is made to a scale of 1 : 100, and its volume is 3 cm³. What is the real building's volume, in m³?
Answer:
9.Two similar triangles have matching sides 3 cm and 4 cm long. The small triangle's area is 90 cm². What is the big one's, in cm²?
Answer:
10.Two similar triangles have matching sides 5 cm and 7 cm long. The small triangle's area is 75 cm². What is the big one's, in cm²?
Answer:

11.Every length of a solid is made 5 times as long (its shape the same). How many times its old surface area is its new one?
- A. 125 times
- B. 25 times
- C. 10 times
- D. 5 times
Choose the right answer
Answer:
12.Two spheres have radii in the ratio 1 : 7, so their volumes are in the ratio 1 : 343.
True or false?
Answer:

13.Two bottles of the same shape are in the ratio 1 : 5 in height. The small one holds 50 mL, the big one mL; their labels' areas are in the ratio : .
Fill in each blank
Working:
Answer:
14.Two statues of the same shape, cast in the same bronze, are 40 cm and 50 cm high. The small one weighs 448 kg. How many kg does the big one weigh?
Working:
Answer:
15.A big cone-shaped sand timer holds 704 cm³ of sand. A smaller one of the same shape is only 14 as high. How many cm³ of sand does it hold?
Working:
Answer:
16.A model car is made to a scale of 1 : 40, and the model has 80 cm² to paint. How many m² does the real car have to paint?
Working:
Answer:
17.Two bottles of the same shape are in the ratio 3 : 5 in height. The small one holds 1350 mL, the big one mL; their labels' areas are in the ratio : .
Fill in each blank
Working:
Answer:
18.Two statues of the same shape, cast in the same bronze, are 10 cm and 30 cm high. The small one weighs 8 kg. How many kg does the big one weigh?
Working:
Answer:

Show the answers and working

Similar figures and solids worksheet: Answers and working

1. 432 cm³
Volumes 1³ : 6³, so the big one = 2 × 6³ ÷ 1³ = 432 cm³.
2. 25, 36, 125, 216
Areas go as the lengths squared, 25 : 36; volumes as cubed, 125 : 216.
3. 54 cm³
Volumes 1³ : 3³, so the big one = 2 × 3³ ÷ 1³ = 54 cm³.
4. 8 cm
The cube root of the volume ratio gives the lengths', 3 : 4: 6 × 4 ÷ 3 = 8 cm.
5. 750 cm³
Areas 4 : 25; the square root gives lengths 2 : 5, and cubing volumes 2³ : 5³: 48 × 5³ ÷ 2³ = 750 cm³.
6. 6 cm
The cube root of the volume ratio gives the lengths', 1 : 2: 3 × 2 ÷ 1 = 6 cm.
7. 162 cm³
Areas 4 : 9; the square root gives lengths 2 : 3, and cubing volumes 2³ : 3³: 48 × 3³ ÷ 2³ = 162 cm³.
8. 3 m³
100 times the lengths is 100³ = 1,000,000 times the volume: 3 × 1,000,000 cm³; 1 m³ = 1 000 000 cm³, so 3 m³.
9. 160 cm²
Similar flat figures' areas also go as the sides squared: 90 × 4² ÷ 3² = 160 cm².
10. 147 cm²
Similar flat figures' areas also go as the sides squared: 75 × 7² ÷ 5² = 147 cm².

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

13. 6250, 1, 25
Capacities 1³ : 5³: 50 × 5³ ÷ 1³ = 6250; areas 1 : 25.
14. 875 kg
Same bronze, so weight goes with volume; volumes go as the heights cubed, 4³ : 5³: 448 × 5³ ÷ 4³ = 875 kg.
15. 11 cm³
Volumes go as (14)³: 704 × 1³ ÷ 4³ = 11 cm³.
16. 12.8 m²
Areas grow 40² = 1600 times: 80 × 1600 cm²; 1 m² = 10 000 cm²: 12.8 m².
17. 6250, 9, 25
Capacities 3³ : 5³: 1350 × 5³ ÷ 3³ = 6250; areas 9 : 25.
18. 216 kg
Same bronze, so weight goes with volume; volumes go as the heights cubed, 1³ : 3³: 8 × 3³ ÷ 1³ = 216 kg.

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