Measures and solids · Ages 13–14

Prisms & cylinders worksheet

Covers: Volume of a prism, Volume of a cylinder, Surface area of a prism, Surface area of a cylinder, Prisms and cylinders at work

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Measures and solids · Ages 13–14

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

1.The prism in the picture has a right-angled triangle as its cross-section, its two shorter sides 12 cm and 6 cm, and it is 15 cm long. What is its volume, in cm³?
Answer:
2.The prism in the picture has an L-shaped cross-section, 7 cm wide and 12 cm high, both arms 3 cm thick, and it is 11 cm long. What is its volume, in cm³?
Answer:
3.Taking π as 3.14, find the volume of the cylinder in the picture: radius 2 cm, height 4 cm. What is it, in cm³?
Answer:
4.The cylinder in the picture has a volume of 243π cm³ and a height of 3 cm. What is its base radius r, in cm?
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5.Cylinder A's radius is 4 times cylinder B's, but A is only 13 as high as B. How many times B's volume is A's?
Answer in simplest form
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6.The prism in the picture has an isosceles triangle as its cross-section: 12 cm across, 8 cm high, its two equal sides 10 cm each; the prism is 7 cm long. What is its total surface area, in cm²?
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7.The cylinder in the picture has a base radius of 10 cm and a height of 3 cm. What is its total surface area, in terms of π?
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8.A cylinder has a base radius of 7 cm and a total surface area of 168π cm². How high is it, in cm?
Answer:
9.The cylindrical container in the picture has a base radius of 4 cm and water 12 cm deep. A stone is dropped in and sinks right under, and the water rises 3 cm. What is the stone's volume, in terms of π?
Answer:

10.The prism in the picture has a triangle 12 cm across and 3 cm high as its cross-section, and is 20 cm long. What is its volume?
- A. (12 + 3) × 20 = 300 cm³
- B. 12 × 3 ÷ 2 + 20 = 38 cm³
- C. 12 × 3 ÷ 2 × 20 = 360 cm³
- D. 12 × 3 × 20 = 720 cm³
Choose the right answer
Answer:
11.Open out the three sides of the prism in the picture and they join into one rectangle, 13 cm wide. How long is the rectangle?
- A. 56 cm
- B. 31 cm
- C. 69 cm
- D. 84 cm
Choose the right answer
Answer:
12.Pour 4 L of water into an empty tank with a base area of 1000 cm², and the water is 4 cm deep.
True or false?
Answer:

13.A model house has a five-sided cross-section: a rectangle 12 cm wide and 4 cm high, with a triangular roof 12 cm across and 4 cm high on top; the house is 6 cm long. The rectangle is cm², the roof cm², and the house's volume cm³.
Fill in each blank
Working:
Answer:
14.The picture shows an L-shaped steel bar: its cross-section 11 cm wide and 10 cm high, both arms 2 cm thick, the bar 21 cm long. Each 1 cm³ of steel weighs 7 g. How many grams does the bar weigh?
Working:
Answer:
15.A can of soda is a cylinder of base radius 3 cm and height 10 cm. Its capacity in terms of π is π cm³; taking π as 3.14, about mL.
Fill in each blank
Working:
Answer:
16.An open-topped cuboid box is to be made from thin board, its bottom 25 cm by 10 cm and its sides 8 cm high. At least how much board does it take?
- A. 1060 cm²
- B. 530 cm²
- C. 780 cm²
- D. 810 cm²
Choose the right answer
Working:
Answer:
17.A big drum is a cylinder 42 cm across and 23 cm high, with skin stretched over its top and bottom and its side painted. In terms of π, how many cm² of skin are there?
Working:
Answer:
18.A cylindrical flask is 5 cm in base radius and 22 cm high inside. Its capacity in terms of π is π cm³; taking π as 3.14, it holds about mL of hot water.
Fill in each blank
Working:
Answer:

Show the answers and working

Prisms & cylinders worksheet: Answers and working

1. 540 cm³
Cross-section 12 × 6 ÷ 2 = 36 cm², volume = cross-section × length = 36 × 15 = 540 cm³.
2. 528 cm³
Split the L into two rectangles: the bottom arm 7 × 3 = 21 and the upright one 9 × 3 = 27, 48 cm²; volume 48 × 11 = 528 cm³.
3. 50.24 cm³
Work in π first: π × 2² × 4 = 16π, then put the value in at the end: 16 × 3.14 = 50.24 cm³.
4. 9 cm
πr² × 3 = 243π, so r² = 243 ÷ 3 = 81 and r = 9 cm (9 × 9 = 81).
5. 163 times
Volume πr²h: 4 times the radius makes r² 16 times; 13 the height, so 16 × 13 = 163 times the volume.
6. 320 cm²
The ends 2 × (12 × 8 ÷ 2) = 96; perimeter 12 + 10 + 10 = 32, sides 32 × 7 = 224; 320 cm² in all.
7. 260 π cm²
Curved 2π × 10 × 3 = 60π, the two ends 2 × π × 10² = 200π, 260π cm² in all.
8. 5 cm
The ends are 2π × 7² = 98π, so the curved surface is 168π − 98π = 70π; height = 70π ÷ (2π × 7) = 70 ÷ 14 = 5 cm.
9. 48 π cm³
The stone's volume equals the water it lifts: base area π × 4² = 16π, times the 3 cm rise: 48π cm³. The water's first depth isn't needed.

10. C. 12 × 3 ÷ 2 × 20 = 360 cm³
The cross-section is 12 × 3 ÷ 2, times the length 20: 360 cm³.
11. A. 56 cm
The three sides laid end to end make the cross-section's perimeter: 7 + 24 + 25 = 56 cm.
12. True
4 L = 4 × 1000 cm³; depth = volume ÷ base area = 4 cm.

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

13. 48, 24, 432
12 × 4 = 48; 12 × 4 ÷ 2 = 24; the cross-section is 72 cm², the volume 72 × 6 = 432 cm³.
14. 5586 g
The cross-section as two rectangles: 11 × 2 + (10 − 2) × 2 = 38 cm²; volume 38 × 21 = 798 cm³; weight 798 × 7 = 5586 g.
15. 90, 282.6
π × 3² × 10 = 90π cm³; 90 × 3.14 = 282.6 cm³ = 282.6 mL (1 cm³ = 1 mL).
16. D. 810 cm²
One bottom, 25 × 10, and four sides, 2 × 10 × 8 + 2 × 25 × 8: 810 cm².
17. 882 π cm²
The skin is two circles: radius 42 ÷ 2 = 21 cm, 2 × π × 21² = 882π cm². The side is the curved surface, not skin: the height isn't needed.
18. 550, 1727
π × 5² × 22 = 550π cm³, 550 × 3.14 = 1727 cm³, which is 1727 mL.

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