Practice Maths

Problem Solving with Volume

Key Ideas

Key Terms

composite solid
A 3D shape made up of two or more simpler solids joined or combined.
add volumes
To find the volume of a composite solid, calculate each part separately and add the results.
subtract volume
To find the volume when a section has been removed, find the total volume and subtract the removed part.
multi-step
A problem requiring more than one calculation step, e.g. convert units, then find volume, then apply a rate.
real-world context
Volume problems set in practical situations such as tanks, pools, packaging, and construction.
unit selection
Choosing the appropriate unit (mm³, cm³, m³, mL, L) to express an answer sensibly.
Hot Tip For composite solids, first decide: are you adding two shapes together or removing a section from a larger one? Drawing a diagram and labelling each part makes the strategy clear.

Worked Example

Composite (add): An L-shaped prism is made of two rectangular prisms: 6×2×4 and 3×2×4 cm.
V1 = 6×2×4 = 48 cm³; V2 = 3×2×4 = 24 cm³; Vtotal = 48 + 24 = 72 cm³

Composite (subtract): A block 10×5×4 cm has a rectangular hole 3×2×4 cm through it.
Vblock = 10×5×4 = 200 cm³; Vhole = 3×2×4 = 24 cm³; Vsolid = 200 − 24 = 176 cm³

Capacity vs Volume

Volume is the amount of three-dimensional space a solid object occupies. Capacity refers to the amount of liquid (or other substance) a container can hold. For practical purposes in metric units, they are equivalent: 1 cm3 = 1 mL, and 1000 cm3 = 1 L.

So if a container has a volume of 8000 cm3, its capacity is 8000 mL = 8 L. Problems that ask "how much water can the tank hold?" are asking for capacity — but you calculate it by finding the volume of the prism and then converting to litres.

Multi-Step Volume Problems

Many real-world volume problems require more than one step. Common multi-step problems include:

  • Finding time to fill a container: Find the volume (or capacity), then divide by the fill rate (e.g. litres per minute).
  • Finding height given volume: Use V = A × h, rearranging to h = V ÷ A.
  • Finding how many items fit in a space: Divide the container's volume by the volume of each item.
  • Mixed units: Convert all measurements to the same unit before calculating.

Always plan your steps before calculating: identify what formula to use, what information you have, what you need to find, and whether units need to be converted.

Composite Prisms

A composite prism (also called a compound prism) is a shape made up of two or more simpler prisms joined together. To find its volume, break it into separate prisms, find the volume of each, and add them together.

Example: An L-shaped prism can be split into two rectangular prisms. Find the volume of each part separately, then add.

Sometimes it is easier to start with a larger, simpler prism and subtract the volume of the cut-out section. Choose whichever approach is simpler for the particular shape.

Real-World Contexts

Volume problems arise constantly in real life:

  • Water tanks: "A cylindrical water tank has diameter 2 m and height 3 m. How many litres does it hold?"
  • Packaging: "How many identical boxes (10 cm × 8 cm × 5 cm) can fit inside a shipping crate of dimensions 1 m × 0.8 m × 0.5 m?"
  • Swimming pools: "A rectangular pool is 12 m long, 4 m wide, and 1.5 m deep. How many kilolitres of water does it hold?"
  • Concrete and materials: "How many cubic metres of concrete are needed to build a rectangular slab 6 m × 4 m × 0.1 m?"

In all these contexts, the key is to identify the correct prism shape, find the cross-sectional area, and multiply by the relevant length or height.

Working Backwards: Finding a Missing Dimension

Sometimes you are given the volume and need to find a missing dimension. Use the formula V = A × h and rearrange:

If you know V and h: A = V ÷ h. If you know V and A: h = V ÷ A.

Example: A rectangular prism has volume 240 cm3, length 8 cm, and width 5 cm. Find the height.
A = 8 × 5 = 40 cm2. h = 240 ÷ 40 = 6 cm.

Key tip: When solving volume problems in real-world contexts, always check your units at the end. If the question asks for litres but your volume is in cm3, divide by 1000 to convert. If it asks for kL, divide by 1 000 000 cm3 (since 1 kL = 1 m3 = 1 000 000 cm3). Writing out your unit conversions explicitly at the end of each problem will prevent errors and show clear working in exams.

Mastery Practice

  1. Find the total volume of each composite solid (two rectangular prisms joined together). Fluency

    1. Prism A: 8×4×3 cm; Prism B: 5×4×3 cm
    2. Prism A: 10×2×6 m; Prism B: 4×2×6 m
    3. Prism A: 7×3×5 cm; Prism B: 7×3×2 cm
    4. Prism A: 12×4×2 m; Prism B: 6×4×3 m
    5. Prism A: 9×5×4 cm; Prism B: 3×5×4 cm
    6. Prism A: 15×3×2 m; Prism B: 5×3×8 m
    7. Prism A: 8×6×3 cm; Prism B: 4×2×3 cm
    8. Prism A: 20×5×4 cm; Prism B: 10×5×2 cm

  2. Find the volume of each solid with a rectangular hole through it. Fluency

    1. Outer block: 10×8×5 cm; Hole: 4×3×5 cm
    2. Outer block: 12×6×4 m; Hole: 3×2×4 m
    3. Outer block: 15×10×6 cm; Hole: 5×4×6 cm
    4. Outer block: 8×8×8 cm (cube); Hole: 2×2×8 cm
    5. Outer block: 20×10×5 m; Hole: 6×4×5 m
    6. Outer block: 18×9×3 cm; Hole: 6×3×3 cm
    7. Outer block: 25×12×8 cm; Hole: 5×4×8 cm
    8. Outer block: 14×10×6 m; Hole: 4×4×6 m

  3. Find the volume of each prism. Fluency

    1. Rectangular prism: 11×7×4 cm
    2. Triangular prism: base = 8 cm, height = 5 cm, length = 9 cm
    3. Rectangular prism: 6×6×10 m
    4. Trapezoidal prism: parallel sides 4 m and 8 m, height 5 m, length 7 m
    5. Triangular prism: base = 12 mm, height = 7 mm, length = 10 mm
    6. Rectangular prism: 3.5×4×6 cm
    7. Triangular prism: base = 9 m, height = 6 m, length = 11 m
    8. Trapezoidal prism: parallel sides 3 cm and 7 cm, height 4 cm, length 8 cm

  4. For each pair, calculate both volumes and state which container holds more. Understanding

    1. Container A: rectangular prism 8×6×5 cm; Container B: rectangular prism 10×4×6 cm
    2. Container A: triangular prism (base 6 m, height 4 m, length 10 m); Container B: rectangular prism 5×4×6 m
    3. Container A: rectangular prism 12×8×3 cm; Container B: triangular prism (base 10 cm, height 8 cm, length 9 cm)
    4. Container A: rectangular 20×15×10 cm; Container B: rectangular 25×12×10 cm
    5. Container A: rectangular prism 9×9×9 cm (cube); Container B: rectangular prism 12×8×7 cm
    6. Container A: trapezoidal prism (parallel sides 5 m and 9 m, height 4 m, length 6 m); Container B: rectangular prism 8×5×6 m
    7. Container A: rectangular prism 50×40×30 cm; Container B: rectangular prism 60×30×35 cm
    8. Container A: triangular prism (base 8 cm, height 6 cm, length 20 cm); Container B: rectangular prism 10×4×12 cm

  5. Solve each multi-step problem. Show all working. Understanding

    1. A rectangular fish tank (60 cm × 30 cm × 40 cm) is filled at a rate of 2 litres per minute. How long does it take to fill the tank completely?
    2. A rectangular storage container (1.2 m × 0.8 m × 0.5 m) is to be filled with small boxes each measuring 20 cm × 10 cm × 10 cm. How many small boxes fit in the container? (Assume perfect packing with no gaps.)
    3. A swimming pool is a rectangular prism 25 m long, 10 m wide, and 1.8 m deep. It is currently half full. How many litres of water are needed to fill it completely?
    4. Water flows into a triangular prism trough (triangle base 50 cm, triangle height 30 cm, length 2 m) at 5 litres per minute. How many minutes will it take to fill the trough?
    5. A rectangular box (30 cm × 20 cm × 15 cm) holds small cubes with 5 cm sides. How many small cubes fit in the box?
    6. A rainwater tank is a rectangular prism 1.5 m × 1 m × 2 m. After rain, it has 1200 L of water. What percentage of the tank is full?
    7. A concrete ramp is a triangular prism: the triangular cross-section has a base of 3 m and a height of 0.4 m, and the ramp is 5 m long. Concrete costs $220 per m³. What is the cost of the ramp?
    8. A rectangular paddling pool (180 cm × 90 cm × 30 cm) is filled using a hose that delivers 9 litres per minute. How long does it take to fill? Give your answer in minutes and seconds.

  6. Solve each problem, showing all working and units at each step. Problem Solving

    1. A manufacturer wants to design an open-top rectangular box with no lid. The base must be 20 cm × 15 cm and the box must hold exactly 3 litres.
      1. What volume in cm³ must the box hold?
      2. What height is required?
      3. How much cardboard (surface area) is needed? (No lid; base + 4 sides.)
    2. A builder is pouring concrete for two steps. Each step is a rectangular prism. Step 1: 1.8 m × 0.3 m × 0.18 m. Step 2 (placed on top of Step 1, at the back): 1.8 m × 0.3 m × 0.18 m.
      1. Calculate the volume of concrete in each step.
      2. Find the total volume of concrete needed in m³.
      3. Concrete is delivered in 0.2 m³ loads. How many loads are needed?
    3. In a science experiment, a student measures the volume of an irregular rock by displacement. A rectangular tank (15 cm × 12 cm × 20 cm) is half filled with water. When the rock is placed in the tank, the water level rises by 2 cm.
      1. What is the volume of the tank in cm³?
      2. By how many cm³ did the water volume increase?
      3. What is the volume of the rock?

  7. Real-world volume problems involving pools and tanks. Problem Solving

    Strategy. Identify the prism type, calculate volume, convert to litres where needed, then apply the context (rates, costs, percentages).
    1. A backyard pool is a rectangular prism 8 m long, 4 m wide, and 1.2 m deep.
      1. Find the volume of the pool in m³.
      2. How many litres does the full pool hold?
      3. The pool is currently 60% full. How many litres need to be added to fill it?
      4. Water is added at 200 L per minute. How long (in minutes) will it take to fill from 60%?
    2. A cylindrical water tank has been approximated as a rectangular prism with dimensions 1 m × 1 m × 2 m. It is currently ¼ full. Water is added at 5 L per minute. How long does it take to fill the tank completely? (Give your answer in minutes.)
    3. A fish pond is a trapezoidal prism: the trapezoidal cross-section has parallel sides of 1.5 m and 2.5 m, a height of 0.6 m. The pond is 3 m long.
      1. Find the volume in m³.
      2. Convert to litres.

  8. Volume problems involving storage and packaging. Problem Solving

    Packing problems. Divide the total volume by the item volume to find how many fit (assuming perfect packing).
    1. A warehouse shelf holds rectangular boxes, each measuring 30 cm × 20 cm × 15 cm. The shelf space is 1.8 m × 0.6 m × 1.5 m. How many boxes fit on the shelf? (No stacking beyond 1.5 m height.)
    2. A rectangular crate (120 cm × 80 cm × 60 cm) is filled with smaller boxes each measuring 20 cm × 10 cm × 15 cm. How many small boxes fit in the crate?
    3. A company needs a box that holds exactly 2.4 L. The box must be 20 cm long and 10 cm wide. What height is needed?
    4. A moving box is 50 cm × 40 cm × 35 cm. Books each occupy a volume of approximately 700 cm³. Estimate how many books fit in the box.

  9. Composite solid problems. Problem Solving

    Composite strategy. Decide whether to add or subtract volumes. Label each part clearly and calculate separately.
    1. A concrete step is made by stacking two rectangular prisms. The lower step is 1.5 m × 0.4 m × 0.2 m. The upper step is 1.5 m × 0.3 m × 0.2 m.
      1. Find the volume of each step.
      2. Find the total volume of concrete needed.
      3. Concrete costs $180 per m³. What is the total cost?
    2. A tunnel is made by removing a rectangular channel (2 m × 1.5 m × 10 m) from a rectangular block of earth (4 m × 3 m × 10 m). Find the volume of earth remaining after the tunnel is cut.
    3. A raised garden bed is an L-shaped prism. Its cross-section consists of two rectangles: 4 m × 0.4 m and 2 m × 0.4 m. The bed is 1.5 m long. Find the total volume of soil needed to fill it.
    4. A decorative block is a rectangular prism (10 cm × 8 cm × 6 cm) with two identical rectangular holes cut through it. Each hole is 2 cm × 2 cm × 6 cm. Find the volume of the solid material.

  10. Extended volume investigations. Problem Solving

    Investigation. These questions require combining multiple volume and conversion skills in a realistic scenario.
    1. A dam is modelled as a trapezoidal prism with parallel sides 20 m and 40 m, height 5 m, and length 100 m.
      1. Calculate the volume of water the dam can hold in m³.
      2. Convert to kilolitres (1 kL = 1 m³).
      3. Water is released at 500 kL per hour. How many hours does it take to empty the dam?
    2. A rectangular swimming pool is 20 m × 10 m. The depth varies: the shallow end is 1 m deep and the deep end is 3 m deep. Model the pool as a trapezoidal prism (parallel depths 1 m and 3 m, length 20 m, width 10 m).
      1. Find the volume of the pool in m³.
      2. How many litres does it hold?
      3. A pump empties the pool at 3000 L per minute. How long does it take to empty? Give your answer in hours and minutes.
    3. A builder pours a concrete path around the outside of a rectangular garden. The garden is 6 m × 4 m. The path is 0.5 m wide on all sides and 0.1 m deep.
      1. Find the outer dimensions of the path area.
      2. Find the volume of the path in m³ (subtract the garden area from the total area, then multiply by depth).
      3. How much does the concrete cost at $220 per m³?
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