Practice Maths

L24 — Calculating Volume of Rectangular Prisms

Key Terms

V = l × w × h
The volume formula for a rectangular prism: multiply length, width, and height. All three dimensions must use the same unit before multiplying.
base area
The area of the face the solid rests on (length × width). Volume = base area × height.
missing dimension
When the volume and two dimensions are known, rearrange the formula to find the third: e.g., h = V ÷ (l × w).
capacity
How much a container holds. Linked to volume: 1 cm³ = 1 mL, and 1 m³ = 1000 L.

The Two Volume Formulas

V = l × w × h — multiply all three dimensions.

V = base area × h — useful when the base area is already known.

Units Matter! All three dimensions must be in the same unit before multiplying. Volume always carries a cubed exponent (cm³, m³, mm³). Never write cm² for a volume answer.

Worked Example

A fish tank is 50 cm long, 25 cm wide, and 30 cm tall. Find its volume.

Step 1 — Formula: V = l × w × h

Step 2 — Substitute: V = 50 × 25 × 30

Step 3 — V = 37 500 cm³

The Formula for Volume of a Rectangular Prism

A rectangular prism (also called a cuboid) has three dimensions: length (l), width (w), and height (h). The volume formula is: V = l × w × h.

This works because you calculate the area of the base (l × w) and then multiply by the height to count all the layers stacked on top of each other.

  • A box 10 cm × 5 cm × 3 cm: V = 10 × 5 × 3 = 150 cm³
  • A room 6 m × 4 m × 2.5 m: V = 6 × 4 × 2.5 = 60 m³
  • A cube with side 4 cm: V = 4 × 4 × 4 = 64 cm³
It does not matter which dimension you call the “length,” “width,” or “height” — multiplication is commutative, so the order never changes the answer. 4 × 5 × 6 = 6 × 5 × 4 = 120 either way. Just use all three dimensions.

Finding a Missing Dimension

If the volume and two dimensions are given, rearrange the formula to find the missing one:

  • V = 120 cm³, l = 6 cm, w = 4 cm. Find h: h = 120 ÷ (6 × 4) = 120 ÷ 24 = 5 cm
  • V = 200 m³, l = 10 m, h = 4 m. Find w: w = 200 ÷ (10 × 4) = 200 ÷ 40 = 5 m

Volume and Capacity: mL and cm³

A very important conversion for everyday life: 1 cm³ = 1 mL. This means volume and capacity are directly linked.

  • Fish tank 30 cm × 20 cm × 25 cm: V = 15 000 cm³ = 15 000 mL = 15 L
  • Juice box 6 cm × 4 cm × 10 cm: V = 240 cm³ = 240 mL

Also: 1 m³ = 1 000 000 cm³ = 1000 L = 1 kilolitre (kL).

Real-Life Volume Problems

Volume questions often ask how much something holds or costs to fill:

  • “How many 1 cm cubes fit in a box 5 cm × 4 cm × 3 cm?” → V = 60 → 60 cubes
  • “How much soil fills a garden bed 2 m × 1.5 m × 0.2 m?” → V = 2 × 1.5 × 0.2 = 0.6 m³
Compound prisms are made of two or more rectangular prisms joined together. Calculate the volume of each part separately, then add the results.

  1. Composite Base Area

    Find the area of the shaded region on the base of this prism, then use it to find the volume.

    20 cm 15 cm Cutout: 10 cm × 4 cm
    1. What is the area of the large outer rectangle?
    2. What is the area of the cutout (inner rectangle)?
    3. What is the area of the shaded base region?
    4. If this shape is the base of a prism 5 cm tall, what is the total volume?

  2. Find the Volume

    Use V = l × w × h to find the volume of each rectangular prism. State your answer with the correct cubic unit.

    5 cm 4 cm 3 cm (a)
    8 m 2 m 3 m (b)
    6.5 cm 3 cm 4 cm (c)
    10 mm 4 mm 5 mm (d)
    6 cm 6 cm 6 cm (e)
    1. Find the volume of prism (a).
    2. Find the volume of prism (b).
    3. Find the volume of prism (c).
    4. Find the volume of prism (d).
    5. The solid in (e) is a cube. Find its volume. Explain why the formula simplifies to V = s³.

  3. Counting the Layers

    A prism is built using 1 cm³ blocks. It is 5 blocks long, 4 blocks wide, and 3 blocks high.

    1. How many blocks are in the bottom layer?
    2. What is the area of the base in cm²?
    3. How many layers are there in total?
    4. What is the total volume of the prism in cm³?

  4. Base Area and Height

    A rectangular box has a base area of 32 cm² and a height of 10 cm.

    1. If the length is 8 cm, what is the width?
    2. Calculate the volume using V = base area × height.
    3. If the height is halved to 5 cm, what is the new volume?
    4. If the base area doubles but the height stays at 10 cm, how does the volume change?

  5. Applying the Formula

    Use the diagram below to calculate the volume.

    10 cm 6 cm 8 cm
    1. Identify the length, width, and height from the diagram.
    2. Calculate the volume of the prism.
    3. What would the volume be if the width were tripled?
    4. What is the area of the top face of this prism?

  6. Special Case: The Cube

    A wooden cube has a side length of 5 cm.

    1. What is the area of one face?
    2. Calculate the volume of the cube.
    3. If the side length is doubled to 10 cm, what is the new volume?
    4. How many of the original 5 cm cubes would fit inside the new 10 cm cube?

  7. Working with Decimals

    A shipping container is 6 m long, 2.5 m wide, and 2 m high.

    1. Calculate the volume in m³.
    2. Convert the length of 6 m into centimetres.
    3. A second container has half the height of this one. What is its volume?
    4. If 1 m³ can hold 1000 litres, what is the capacity of the original container in litres?

  8. Finding the Missing Dimension

    Each diagram shows a rectangular prism with one unknown dimension. Use the given information to find the missing value.

    h = ? A = 40 cm² V = 200 cm³ (a)
    h = ? 7 cm 3 cm V = 84 cm³ (b)
    V = 27 cm³ s = ? (c)
    10 cm l = ? w = ? V = 150 cm³ (d)
    1. Prism (a) has a base area of 40 cm² and a volume of 200 cm³. Find the height.
    2. Prism (b) has volume 84 cm³, length 7 cm, and width 3 cm. Find the height.
    3. Solid (c) is a cube with volume 27 cm³. Find the side length.
    4. Prism (d) has height 10 cm and volume 150 cm³. Give one possible set of values for the length and width.

  9. The Aquarium

    An aquarium is 50 cm long, 30 cm wide, and 40 cm high. It is currently filled with water to a height of 25 cm.

    1. What is the total volume of the aquarium in cm³?
    2. What is the volume of water currently in the tank?
    3. How much more water (in cm³) is needed to fill the tank completely?
    4. Convert the volume of water currently in the tank to litres.

  10. Compound Prisms

    A concrete step is made of two rectangular blocks. The lower block is 60 cm × 30 cm × 20 cm. The upper block is 30 cm × 30 cm × 20 cm.

    30 cm 20 cm 20 cm 60 cm 30 cm
    1. Calculate the volume of the lower block.
    2. Calculate the volume of the upper block.
    3. What is the total volume of the concrete step?
    4. If 1 cm³ of concrete costs $0.05, how much does the step cost to build?
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