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PSLE Mathematics · Primary 5 and 6

Volume of Cubes and Cuboids

Volume is the amount of space a solid takes up or a container holds. In upper primary you find the volume of cubes and cuboids, convert between cubic centimetres and litres, find an unknown length from the volume, and solve problems about water in rectangular tanks. Tank questions often combine volume with fractions, decimals and rates of filling, so they reward neat working. This guide explains the formulas, the link between litres and cm³, water level problems and a full worked example.

What the syllabus expects

  • Find the volume of a cube or cuboid by counting unit cubes and by using length × breadth × height
  • Use cubic units such as cm³ and m³
  • Convert between litres, millilitres and cubic centimetres, using 1 ℓ = 1000 cm³
  • Find an unknown edge of a cube or cuboid given its volume and other measurements
  • Find the volume of liquid in a rectangular tank and the height of the liquid
  • Solve problems where water is added, removed or an object is placed in a tank

Volume of cubes and cuboids

Volume is measured by counting how many unit cubes fit inside a solid. A 1 cm cube has a volume of 1 cm³. A cuboid that is 8 cm long, 5 cm wide and 3 cm high holds 8 × 5 = 40 cubes in each layer, and there are 3 layers, so its volume is 40 × 3 = 120 cm³. This gives the formula volume of cuboid = length × breadth × height. A cube has all its edges equal, so volume of cube = edge × edge × edge. A cube with 4 cm edges has a volume of 4 × 4 × 4 = 64 cm³. Another useful form is volume = base area × height, because length × breadth is the area of the base. Larger solids may be measured in m³, where 1 m³ is a cube with 1 m edges.

Litres and cubic centimetres

The capacity of a container is the volume of liquid it can hold. The key link is 1 ℓ = 1000 cm³, and so 1 ml = 1 cm³. A tank 50 cm long, 30 cm wide and 40 cm high holds 50 × 30 × 40 = 60 000 cm³ of water when full, which is 60 000 ÷ 1000 = 60 ℓ. To change litres into cm³, multiply by 1000: 2.5 ℓ = 2500 cm³. When a tank is only partly filled, the water itself forms a cuboid with the same base as the tank. Its volume is length × breadth × height of the water, not the height of the tank. For example, if that tank is filled to a height of 25 cm, it holds 50 × 30 × 25 = 37 500 cm³ = 37.5 ℓ.

Finding an unknown length

If you know the volume and two of the measurements, divide to find the third. Height = volume ÷ (length × breadth), which is volume ÷ base area. A cuboid with a volume of 360 cm³ and a base of 9 cm by 8 cm has a height of 360 ÷ 72 = 5 cm. For a cube, you need the number that, multiplied by itself three times, gives the volume. A cube of volume 125 cm³ has edges of 5 cm, because 5 × 5 × 5 = 125. Learning the cubes 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000 makes these questions quick. Similarly, if a cube's face has an area of 36 cm², its edge is 6 cm, because 6 × 6 = 36.

Water level problems

When water is poured into or out of a rectangular tank, the change in volume equals the base area × the change in height. A tank with a base of 40 cm by 25 cm has a base area of 1000 cm², so pouring in 3 ℓ = 3000 cm³ raises the water level by 3000 ÷ 1000 = 3 cm. When a solid object is placed in the water and is completely covered, the water level rises, and the volume of the object equals the base area × the rise in water level. Water moved from one tank to another keeps the same volume, so find the volume in cm³ first, then divide by the new tank's base area. Always check whether the question asks for the new height of the water or the change in height.

Worked example

A rectangular tank measures 60 cm by 40 cm by 50 cm (height). It is 3/5 filled with water. Then 18 ℓ of water is poured out. (a) How many litres of water were in the tank at first? (b) What is the height of the water after 18 ℓ is poured out?

  1. Height of water at first = 3/5 × 50 = 30 cm.
  2. Volume of water at first = 60 × 40 × 30 = 72 000 cm³.
  3. Convert using 1 ℓ = 1000 cm³: 72 000 cm³ = 72 ℓ.
  4. Water poured out = 18 ℓ = 18 000 cm³.
  5. Water left = 72 000 − 18 000 = 54 000 cm³.
  6. Base area of the tank = 60 × 40 = 2400 cm².
  7. New height = 54 000 ÷ 2400 = 22.5 cm.
  8. Check: the drop in height is 18 000 ÷ 2400 = 7.5 cm, and 30 − 7.5 = 22.5 cm.

Answer: (a) 72 ℓ (b) 22.5 cm

Common mistakes

  • Mixing litres and cm³ in one calculation. Change everything into cm³ using 1 ℓ = 1000 cm³ before you calculate, then convert back if the answer is wanted in litres.
  • Using the height of the tank instead of the height of the water. Only the water forms the cuboid you need.
  • Dividing by the wrong pair of measurements. To find the water height, divide the volume by the base area (length × breadth), not by length × height.
  • Dividing a cube's volume by 3 to find its edge. A cube of volume 27 cm³ has edges of 3 cm because 3 × 3 × 3 = 27, not because 27 ÷ 3 = 9.
  • Treating the rise in water level as the volume of an object. The volume of the object is the base area × the rise, and this only works when the object is fully under water.
  • Writing cm² for a volume. Volume is in cubic units such as cm³ or m³.

Exam tips

  • Sketch the tank, mark the water level, and label the base area before you calculate. Most tank questions then become one multiplication or division.
  • Convert litres into cm³ at the start of the question and write units on every line.
  • Learn the cubes of 1 to 10 so you can find a cube's edge from its volume without trial and error.

Practice questions

1. Easy: Find the volume of a cuboid that is 12 cm long, 5 cm wide and 4 cm high.

Show answer

240 cm³. Working: 12 × 5 × 4 = 60 × 4 = 240 cm³.

2. Medium: A cube has a volume of 343 cm³. Find the length of one edge and the area of one face.

Show answer

Edge 7 cm, face area 49 cm². Working: 7 × 7 × 7 = 343, so each edge is 7 cm. One face has area 7 × 7 = 49 cm².

3. Harder: A rectangular tank measures 40 cm by 25 cm by 30 cm (height). It contains water to a height of 12 cm. A stone is placed in the tank and is completely covered, and the water level rises to 15 cm. (a) Find the volume of the stone. (b) How many more litres of water are needed to fill the tank to the brim?

Show answer

(a) 3000 cm³ (b) 15 ℓ. Working: base area = 40 × 25 = 1000 cm². The rise is 15 − 12 = 3 cm, so the stone's volume is 1000 × 3 = 3000 cm³. The empty space above the water is 30 − 15 = 15 cm high, so it holds 1000 × 15 = 15 000 cm³ = 15 ℓ.

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