Secondary 3 · Mathematics · Measurement
Surface area and volume of prisms and cylinders
Calculate the surface area and volume of right prisms and cylinders, and solve practical problems such as painting or filling containers.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- Calculate the surface area of right prisms by adding the areas of their faces.
- Calculate the surface area and volume of cylinders.
- Use volume = area of cross-section × length, and apply surface area and volume to practical problems.
Success looks like
- I can sketch the net of a prism and add the areas of all the faces.
- I can use SA = 2πr² + 2πrh and V = πr²h for a cylinder.
- I can decide whether a problem needs surface area (painting, wrapping) or volume (filling), and convert cm³ or m³ to litres.
The big idea
The surface area of a solid is the total area of all its faces. A good method is to imagine (or draw) its net, find the area of each face, and add them up. Surface area is measured in square units (cm², m²).
For a rectangular prism with length l, width w and height h, there are three pairs of matching faces, so SA = 2(lw + lh + wh).
For any right prism, the surface area is the two ends plus the rectangles around the sides. The rectangles together form one long rectangle whose length is the perimeter of the cross-section, so:
- SA = 2 × (area of cross-section) + (perimeter of cross-section) × length.
A cylinder has two circular ends and a curved surface. If you unroll the curved surface it becomes a rectangle: its width is the circumference of the circle (2πr) and its height is h. So:
- SA of a closed cylinder = 2πr² + 2πrh.
- For an open cylinder (no lid), leave out one circle: πr² + 2πrh.
The volume of any prism or cylinder is the area of the cross-section times the length (or height): V = A × h. For a cylinder, the cross-section is a circle, so V = πr²h. Volume is measured in cubic units (cm³, m³).
In practical problems, decide what is being asked. Painting, wrapping or making a container uses surface area. Filling a container uses volume and capacity: 1 cm³ = 1 mL, 1000 cm³ = 1 L and 1 m³ = 1000 L. Use the π button on your calculator and round only at the end.
Worked examples
Rectangular prism
Find the surface area and volume of a box 5 cm long, 4 cm wide and 3 cm high.
- Pairs of faces: 5 × 4 = 20, 5 × 3 = 15 and 4 × 3 = 12.
- SA = 2(20 + 15 + 12) = 2 × 47 = 94 cm².
- V = 5 × 4 × 3 = 60 cm³.
Answer: SA = 94 cm², V = 60 cm³
Triangular prism
A prism has a right-angled triangle cross-section with sides 3 cm, 4 cm and 5 cm, and is 10 cm long. Find its surface area and volume.
- Area of the triangle = ½ × 3 × 4 = 6 cm². There are two ends: 2 × 6 = 12 cm².
- Perimeter of the triangle = 3 + 4 + 5 = 12 cm. Rectangular faces = 12 × 10 = 120 cm².
- SA = 12 + 120 = 132 cm².
- V = area of cross-section × length = 6 × 10 = 60 cm³.
Answer: SA = 132 cm², V = 60 cm³
Cylinder
A closed cylinder has radius 3 cm and height 10 cm. Find its surface area and volume to 1 decimal place.
- Two circles: 2 × π × 3² = 18π.
- Curved surface: 2 × π × 3 × 10 = 60π.
- SA = 18π + 60π = 78π ≈ 245.0 cm².
- V = π × 3² × 10 = 90π ≈ 282.7 cm³.
Answer: SA ≈ 245.0 cm², V ≈ 282.7 cm³
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Which formula gives the volume of a cylinder?
πr²h — Volume = area of the circular cross-section (πr²) × height.
What is the surface area of a cube with sides of 4 cm?
96 cm² — A cube has 6 faces, each 4 × 4 = 16 cm². 6 × 16 = 96 cm². (64 is the volume in cm³.)
A prism has a cross-section area of 15 cm² and a length of 8 cm. What is its volume?
120 cm³ — V = 15 × 8 = 120 cm³. Volume uses cubic units.
Which formula gives the surface area of a cylinder with no lid?
πr² + 2πrh — There is only one circle (the base) plus the curved surface: πr² + 2πrh.
Find the surface area of a rectangular prism 6 cm long, 2 cm wide and 5 cm high.
104 cm² — 2(6 × 2 + 6 × 5 + 2 × 5) = 2(12 + 30 + 10) = 2 × 52 = 104 cm².
Find the volume of a cylinder with radius 5 cm and height 8 cm, correct to 1 decimal place. Use the π button.
628.3 cm³ — V = π × 5² × 8 = 200π ≈ 628.3 cm³.
Find the curved surface area of a cylinder with radius 2 cm and height 7 cm, correct to 1 decimal place.
88.0 cm² — Curved surface = 2πrh = 2 × π × 2 × 7 = 28π ≈ 87.96, which is 88.0 cm² to 1 decimal place.
A cylindrical water tank has radius 0.5 m and height 1.2 m. How many litres does it hold, to the nearest litre?
942 L — V = π × 0.5² × 1.2 = 0.3π ≈ 0.9425 m³. 1 m³ = 1000 L, so it holds about 942 L.
Watch out for
- Mixing up surface area and volume, or their units. Surface area covers the outside (cm²); volume fills the inside (cm³).
- Using the diameter instead of the radius in πr². If the diameter is given, halve it first.
- Forgetting faces, such as one of the two ends of a prism, or including a lid that isn't there. Sketching the net and ticking off each face prevents this.
Tips for parents
- Open out a cereal box to see its net, measure it, and check that the learner's calculated surface area matches.
- Measure a can, calculate its volume in cm³, and compare with the millilitres printed on the label (1 cm³ = 1 mL).
- Encourage the learner to keep π in their working (such as 78π) and round only at the final step.
Go further
- Design a can that holds 375 mL using the least metal. Try different radii and work out the height and surface area of each.
- Estimate how much paint is needed for the walls of a room, given that one litre covers about 12 m².
- Work out how many litres a rectangular garden bed or fish tank holds, and how long it would take to fill with a hose.
About 55 minutes
You'll need
- Calculator with a π button
- An empty cereal box and a can or cardboard tube
- Ruler or tape measure, scissors
Key words
- Right prism
- A solid with two identical parallel ends (the cross-section) joined by rectangular faces at right angles to the ends.
- Cross-section
- The shape you see when you slice a solid parallel to its end. For a prism it is the same all the way along.
- Net
- A flat pattern that folds up to make a 3D solid.
- Surface area (SA)
- The total area of all the outside surfaces of a solid, measured in square units such as cm².
- Volume
- The amount of space a solid takes up, measured in cubic units such as cm³.
- Capacity
- How much a container holds, often in litres. 1 L = 1000 cm³ and 1 m³ = 1000 L.
Still tricky?
For parents
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