Teach Me Maths

Volume of Cuboids and Prisms

Question 11 mark

The diagram shows a cube. Work out its volume, in cubic centimetres.

Hint

Multiply the side length by itself three times.

Worked solution

The diagram shows a cube. Work out its volume, in cubic centimetres.

  1. V=3\times3\times3=27\text{ cm}^3.

Question 22 marks

Work out the volume of the cuboid, in cubic centimetres.

Hint

Find the area of the base, then multiply by the height.

Worked solution

Work out the volume of the cuboid, in cubic centimetres.

  1. Base area: 8\times3=24\text{ cm}^2.
  2. V=24\times4=96\text{ cm}^3.

Question 32 marks

Work out the volume of the cuboid, in cubic metres.

Hint

All three dimensions are already in metres.

Worked solution

Work out the volume of the cuboid, in cubic metres.

  1. V=5\times2\times1.5=15\text{ m}^3.
  2. Multiplying three lengths gives a volume.

Question 41 mark

A prism has cross-sectional area 18\text{ cm}^2 and length 7 cm. Find its volume in cubic centimetres.

Hint

Use cross-sectional area times length.

Worked solution

A prism has cross-sectional area 18\text{ cm}^2 and length 7 cm. Find its volume in cubic centimetres.

  1. V=18\times7=126\text{ cm}^3.

Question 53 marks

Work out the volume of the triangular prism, in cubic centimetres.

Hint

Find the triangular area first, then multiply by the prism length.

Worked solution

Work out the volume of the triangular prism, in cubic centimetres.

  1. Cross-sectional area: \frac12\times6\times4=12\text{ cm}^2.
  2. V=12\times10=120\text{ cm}^3.

Question 62 marks

The cuboid has volume 120\text{ cm}^3. Work out the height h, in centimetres.

Hint

Find the base area, then divide the volume by it.

Worked solution

The cuboid has volume 120\text{ cm}^3. Work out the height h, in centimetres.

  1. Base area: 5\times4=20\text{ cm}^2.
  2. Height: 120\div20=6 cm.

Question 72 marks

A prism has volume 144\text{ cm}^3 and length 12 cm. Find its cross-sectional area in square centimetres.

Hint

Undo multiplying by the prism length.

Worked solution

A prism has volume 144\text{ cm}^3 and length 12 cm. Find its cross-sectional area in square centimetres.

  1. A\times12=144.
  2. A=144\div12=12\text{ cm}^2.

Question 81 mark

A container holds 3500\text{ cm}^3 of water. How many litres is this? Use 1000\text{ cm}^3=1 litre.

Hint

Divide the cubic centimetres by the number in one litre.

Worked solution

A container holds 3500\text{ cm}^3 of water. How many litres is this? Use 1000\text{ cm}^3=1 litre.

  1. 3500\div1000=3.5 litres.

Question 91 mark

The diagram shows a cube. Which calculation finds its volume?

Hint

Volume counts layers of cubes, not the area of the six faces.

Worked solution

The diagram shows a cube. Which calculation finds its volume?

  1. 4\times3
  2. 4\times4\times4
  3. 6\times4^2
  4. 4+4+4
  1. V=4\times4\times4=64\text{ cm}^3.
  2. 6\times4^2 would calculate the total area of the faces instead.

Question 103 marks

Work out the volume of the triangular prism, in cubic centimetres.

Hint

Use the perpendicular sides to find the triangular end’s area; the sloping 10 cm side is not its height.

Worked solution

Work out the volume of the triangular prism, in cubic centimetres.

  1. End area: \frac12\times6\times8=24\text{ cm}^2.
  2. Volume: 24\times5=120\text{ cm}^3.

Question 113 marks

The diagram gives the inside dimensions of a rectangular fish tank. Work out its full capacity, in litres. Use 1000\text{ cm}^3=1 litre.

Hint

Find the volume in cubic centimetres, then convert it to litres.

Worked solution

The diagram gives the inside dimensions of a rectangular fish tank. Work out its full capacity, in litres. Use 1000\text{ cm}^3=1 litre.

  1. V=50\times30\times40=60000\text{ cm}^3.
  2. 60000\div1000=60 litres.

Question 123 marks

The triangular prism has volume 240\text{ cm}^3. Work out the length x, in centimetres.

Hint

Find the triangular area, then divide the volume by this area.

Worked solution

The triangular prism has volume 240\text{ cm}^3. Work out the length x, in centimetres.

  1. Cross-sectional area: \frac12\times8\times5=20\text{ cm}^2.
  2. Length: 240\div20=12 cm.

Question 133 marks

Work out the volume of the prism, in cubic centimetres.

Hint

Find the trapezium’s area before using the prism length.

Worked solution

Work out the volume of the prism, in cubic centimetres.

  1. Cross-sectional area: \frac12(4+10)\times3=21\text{ cm}^2.
  2. Volume: 21\times8=168\text{ cm}^3.

Question 144 marks

Small cartons are packed into a storage box. The diagrams show the outside dimensions of a carton and the inside dimensions of the box. Work out the greatest number of cartons that can fit in the box.

Hint

Use the same units for the cartons and box. Find how many fit along each dimension, then multiply.

Worked solution

Small cartons are packed into a storage box. The diagrams show the outside dimensions of a carton and the inside dimensions of the box. Work out the greatest number of cartons that can fit in the box.

  1. The carton dimensions in centimetres are 5\times4\times3.
  2. Place the 5 cm edges along the 40 cm side: 40\div5=8 cartons. Along the other sides, 24\div4=6 and 18\div3=6 cartons fit.
  3. This arrangement fits 8\times6\times6=288 cartons, with no gaps.
  4. The box volume divided by the carton volume is \frac{40\times24\times18}{5\times4\times3}=288, so no arrangement can hold more.

Question 154 marks

The diagram shows the cross-section of a prism. The prism is 10 cm long. Work out its volume, in cubic centimetres.

Hint

Subtract the missing area from the full rectangle, then multiply by the prism length.

Worked solution

The diagram shows the cross-section of a prism. The prism is 10 cm long. Work out its volume, in cubic centimetres.

  1. Complete the cross-section to a 8\times6 rectangle. The hatched upper-right corner is missing: it is 3\times4.
    Complete the rectangle, then subtract the cornerThe full rectangle measures 8 by 6 cm. The orange hatched upper-right corner measures 3 by 4 cm and is missing from the blue L shape.8 cm6 cm3 cm4 cmSubtract the hatched corner
    The dashed edges complete the whole rectangle. Subtract the hatched corner to leave the L shape.
  2. Whole rectangle: 8\times6=48\text{ cm}^2. Missing corner: 3\times4=12\text{ cm}^2.
  3. Cross-section area: 48-12=36\text{ cm}^2.
  4. Volume: 36\times10=360\text{ cm}^3.

Question 16Challenge4 marks

The diagram gives the inside dimensions of a rectangular container. It is \frac34 full of juice. Each cup holds 250 ml. Work out the greatest number of complete cups that can be filled from the container. Use 1\text{ cm}^3=1 ml.

Hint

Find the volume of juice, then divide by the volume of one cup. Only completely filled cups count.

Worked solution

The diagram gives the inside dimensions of a rectangular container. It is \frac34 full of juice. Each cup holds 250 ml. Work out the greatest number of complete cups that can be filled from the container. Use 1\text{ cm}^3=1 ml.

  1. The container capacity is 24\times15\times20=7200\text{ cm}^3=7200 ml.
  2. The juice volume is \frac34\times7200=5400 ml.
  3. 5400\div250=21.6. There is enough juice for 21 complete cups; 22 complete cups would need 5500 ml.
  4. The greatest number of complete cups is 21.

Question 17Challenge4 marks

Six identical metal cubes are melted and recast into a solid cuboid. The diagrams show one cube and the new cuboid. No metal is lost. Work out the height h of the cuboid, in centimetres.

Hint

The total volume stays the same. Find the volume of all six cubes before dividing by the cuboid’s base area.

Worked solution

Six identical metal cubes are melted and recast into a solid cuboid. The diagrams show one cube and the new cuboid. No metal is lost. Work out the height h of the cuboid, in centimetres.

  1. One cube has volume 4^3=64\text{ cm}^3. Six have volume 6\times64=384\text{ cm}^3.
  2. The new base area is 8\times6=48\text{ cm}^2.
  3. Height: 384\div48=8 cm.

Question 18Challenge4 marks

The diagram shows a concrete ramp in the shape of a triangular prism. One bag of mix makes 0.025\text{ m}^3 of concrete. How many bags are needed to make the ramp? Ignore waste.

Hint

Use the triangle’s area times the width to find the concrete volume, then divide by the volume from one bag.

Worked solution

The diagram shows a concrete ramp in the shape of a triangular prism. One bag of mix makes 0.025\text{ m}^3 of concrete. How many bags are needed to make the ramp? Ignore waste.

  1. Triangular area: \frac12\times1.5\times0.4=0.3\text{ m}^2.
  2. Ramp volume: 0.3\times2=0.6\text{ m}^3.
  3. Bags: 0.6\div0.025=24. Exactly 24 bags are needed.

Question 19Challenge5 marks

Work out the volume of the triangular prism to 3 significant figures, in cubic centimetres.

Hint

Find the missing side of the triangular end using its marked right angle. Then find the end area and multiply by the prism length.

Worked solution

Work out the volume of the triangular prism to 3 significant figures, in cubic centimetres.

  1. Let the unlabelled side of the triangular end be b cm. The sloping 11 cm side is opposite the right angle.
  2. b^2=11^2-7^2=121-49=72, so b=\sqrt{72} cm.
  3. The end area is \frac12\times7\times\sqrt{72}\text{ cm}^2.
  4. The prism volume is \frac12\times7\times\sqrt{72}\times15=445.4772\ldots\text{ cm}^3.
  5. To 3 significant figures the volume is 445\text{ cm}^3. Keep the full square-root value until the final rounding.

Question 20Challenge5 marks

The diagram shows the cross-section of a wooden beam. The beam is a prism 120 cm long. Wood costs £2500 per cubic metre. Work out the cost of the wood in this beam, in pounds. Use 1\text{ m}^3=1000000\text{ cm}^3.

Hint

Find the remaining cross-sectional area and the volume, convert the volume to cubic metres, then apply the price.

Worked solution

The diagram shows the cross-section of a wooden beam. The beam is a prism 120 cm long. Wood costs £2500 per cubic metre. Work out the cost of the wood in this beam, in pounds. Use 1\text{ m}^3=1000000\text{ cm}^3.

  1. Complete the cross-section to a 10\times8 rectangle. The hatched upper-right corner is missing: it is 4\times3.
    Complete the rectangle, then subtract the cornerThe full rectangle measures 10 by 8 cm. The orange hatched upper-right corner measures 4 by 3 cm and is missing from the blue L shape.10 cm8 cm4 cm3 cmSubtract the hatched corner
    The dashed edges complete the whole rectangle. Subtract the hatched corner to leave the L shape.
  2. Whole rectangle: 10\times8=80\text{ cm}^2. Missing corner: 4\times3=12\text{ cm}^2.
  3. Cross-section area: 80-12=68\text{ cm}^2.
  4. Volume: 68\times120=8160\text{ cm}^3.
  5. In cubic metres: 8160\div1000000=0.00816\text{ m}^3.
  6. Cost: 0.00816\times2500=20.40, so the wood costs £20.40.

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