Practice Maths

Surface Area and Volume Problem Solving

Key Ideas

Key Terms

Select the correct formula
By first identifying the shape (rectangular prism, triangular prism, cylinder, composite).
Real-world contexts
For surface area: painting, tiling, wrapping, manufacturing packaging.
Real-world contexts
For volume: filling tanks, capacity of containers, amount of material.
Hot Tip Read problem-solving questions carefully. Look for key words: “covers” or “wraps” → surface area; “holds”, “fills”, or “capacity” → volume. Make sure all units match before substituting into formulas.

Surface Area vs Volume: Which Do You Need?

The most important skill in multi-step problems is deciding which measurement to calculate. A useful rule of thumb:

  • Use surface area when you deal with the outside of a solid — painting, wrapping, covering, coating, tiling, insulating.
  • Use volume when you deal with the inside of a solid — filling, holding, containing, pouring, packing with liquid or material.

Common real-world examples: painting a shed (SA), tiling a pool floor (SA), filling a pool with water (V), buying soil for a raised garden bed (V), wrapping a present (SA).

Multi-Step Problem Strategies

Many problems require more than one calculation. A structured approach prevents errors:

  1. Read the problem carefully and identify what you are ultimately asked to find.
  2. Draw a labelled diagram of the solid(s) involved.
  3. Identify which formula(s) you need and write them out.
  4. Substitute all known values and calculate.
  5. Convert units if required (cm to m, cm3 to litres, etc.).
  6. Interpret your answer in context — does it make sense? Round appropriately.

Working with Different Units

Mixing units is a very common source of errors. Before substituting, make sure all measurements are in the same unit. Key conversions:

  • Lengths: 1 m = 100 cm; 1 km = 1000 m
  • Areas: 1 m2 = 10 000 cm2
  • Volumes: 1 m3 = 1 000 000 cm3; 1 L = 1000 cm3

If a room is 4.5 m long and a tile is 30 cm wide, convert everything to metres (or everything to cm) before calculating area.

Real-World Application Examples

Painting walls: Find the total wall area (SA minus floor and ceiling, or just walls), subtract door/window areas, then divide by the coverage rate (e.g., 12 m2 per litre) to find litres of paint needed. Round up — you cannot buy part of a tin.

Filling a pool: Calculate the volume of the pool, convert to litres, and that gives you the water needed. You might also be asked how long filling takes given a pump rate in litres per minute.

Packaging design: Find the surface area of a box to determine the material cost, or compare two box designs to see which uses less cardboard.

Exam tip: When a question asks "how many tins of paint?" or "how many bags of mulch?", always round up to the next whole number — you cannot buy a fraction of a tin. Similarly, when cutting tiles, round up because you need whole tiles even if some are partially used. State your rounding reasoning in your working.

Mastery Practice

  1. For each situation, state whether you need surface area or volume, and name the formula you would use. Fluency

    1. Finding how much paint is needed to coat the outside of a cylindrical water tank.
    2. Finding how much water a rectangular swimming pool holds.
    3. Finding how much cardboard is needed to make a cereal box (rectangular prism, no overlaps).
    4. Finding how much concrete is needed to fill a cylindrical post hole.
    5. Finding the area of aluminium foil needed to wrap a chocolate bar (rectangular prism shape).
    6. Finding the amount of mulch to fill a triangular garden bed (triangular prism cross-section).

  2. A company makes cylindrical tin cans. Each can has a diameter of 10 cm and a height of 14 cm. Understanding

    1. Calculate the total surface area of one can (including both circular ends) to 2 d.p.
    2. The metal for the can costs $0.0045 per cm². Find the material cost per can to the nearest cent.
    3. How many cans can be made from a sheet of metal that is 2 m × 1.5 m? (Assume no waste and that you divide total sheet area by SA per can.)
    4. Calculate the volume of one can in cm³ and in millilitres.
    5. If the cans are filled with soup and soup costs $0.80 per litre, find the cost of soup per can.

  3. A rectangular room is 6 m long, 4.5 m wide, and 2.7 m high. The room has two windows (each 1.2 m × 1.4 m) and one door (0.9 m × 2.1 m) that are not painted. Understanding

    1. Calculate the total wall area (4 walls) before subtracting openings.
    2. Calculate the total area of windows and door.
    3. Find the net wall area to be painted.
    4. The ceiling is also painted. Find the area of the ceiling.
    5. Total paint area = walls + ceiling. One tin of paint covers 12 m² and costs $34. How many tins are needed and what is the total cost? (Apply two coats — double the paint area.)

  4. Two designs are proposed for packaging 600 cm³ of breakfast cereal. Design A is a rectangular box 10 cm × 6 cm × 10 cm. Design B is a cylinder with r = 4.5 cm and h adjusted so the volume equals 600 cm³. Understanding

    1. Verify that Design A has a volume of 600 cm³.
    2. Find the required height of the cylinder in Design B (to 2 d.p.).
    3. Calculate the surface area of Design A.
    4. Calculate the surface area of Design B (using the height found in part b), to 2 d.p.
    5. Which design uses less material? By how much (to 1 d.p.)?

  5. A concrete garden path runs around the outside of a rectangular lawn area 12 m long and 8 m wide. The path is 1 m wide and 0.12 m deep. Problem Solving

    1. Find the outer dimensions of the path-plus-lawn rectangle.
    2. Find the volume of the total outer rectangle (path + lawn area) to depth 0.12 m.
    3. Find the volume of the lawn area only to depth 0.12 m.
    4. Hence find the volume of concrete needed for the path.
    5. Concrete costs $320 per m³. Find the total cost of concrete.

  6. A farmer needs to store water in a cylindrical tank. The tank must hold at least 50 000 litres of water. Two designs are available. Problem Solving

    Design X: radius = 2 m, height = 4 m   |   Design Y: radius = 3 m, height = 2 m

    1. Calculate the volume of Design X in m³ and convert to litres (to the nearest litre).
    2. Calculate the volume of Design Y in m³ and convert to litres (to the nearest litre).
    3. Which design(s) meet the 50 000 litre requirement?
    4. Calculate the total surface area of each design (to 2 d.p.).
    5. The tank is made of steel costing $85 per m². Which design is cheaper to build, and by how much (to the nearest dollar)?

  7. A gift box is a rectangular prism 25 cm × 18 cm × 10 cm. Wrapping paper costs $4.50 per sheet, where each sheet is 70 cm × 55 cm. Allow 15% extra paper for overlaps. Problem Solving

    1. Find the total surface area of the gift box.
    2. Calculate 115% of the SA to account for overlaps.
    3. Find the area of one sheet of wrapping paper.
    4. How many sheets are needed? (Round up to the nearest whole sheet.)
    5. Find the total cost of wrapping paper for 50 gift boxes.

  8. A grain silo consists of a cylinder (r = 3 m, h = 8 m) with a flat base and an open top. The silo is filled with grain to a height of 6 m. Problem Solving

    1. Find the total volume of the silo (full capacity) to 2 d.p.
    2. Find the volume of grain currently in the silo.
    3. What percentage of the silo is currently filled (to 1 d.p.)?
    4. Grain has a density of approximately 800 kg/m³. Find the mass of grain in the silo in tonnes (1 tonne = 1000 kg).
    5. The curved wall of the silo is to be painted with a weatherproof sealant costing $12 per m² (only the curved side is painted, not the base). Find the total cost.

  9. An above-ground swimming pool is in the shape of a cylinder with diameter 4.8 m and depth 1.2 m. A plastic liner covers the curved wall and the base only (the top is open). Problem Solving

    1. Find the area of the base (circular) of the pool.
    2. Find the area of the curved wall.
    3. Find the total area of liner needed (base + curved wall) to 2 d.p.
    4. Liner material costs $8.50 per m². Find the cost of the liner.
    5. How many litres of water does the pool hold when filled to 90% of its depth?

  10. A chocolate bar is in the shape of a triangular prism. The triangular cross-section is a right triangle with legs 3 cm and 4 cm (hypotenuse 5 cm). The bar is 15 cm long. Problem Solving

    1. Find the volume of the chocolate bar.
    2. Chocolate has a density of 1.3 g/cm³. Find the mass of the bar in grams.
    3. Find the total surface area of the chocolate bar.
    4. Foil packaging covers the entire surface. If foil costs $0.002 per cm², find the cost to package one bar.
    5. In a production run of 500 bars, what is the total volume of chocolate used (in litres)? Recall 1 L = 1000 cm³.
See Answers ➔