Step-by-step worked examples and graded practice questions on volume and surface area — cuboids, prisms, cylinders, cones, spheres and compound 3D shapes.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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Volume = l × w × h. Surface area = the sum of the areas of all 6 faces.
Worked Example 2 — Prism
A triangular prism has a cross-sectional area of 12cm² and a length of 9cm. Find its volume.
1
Volume of a prism = cross-sectional area × length
2
Volume = 12 × 9 = 108cm³
Answer108cm³
A prism has the same cross-section all the way along its length.
Cylinders
Volume of a cylinder = πr²h
Surface area of a cylinder = 2πr² + 2πrh (two circular ends, plus the curved side)
Worked Example 3 — Cylinder volume
Find the volume of a cylinder with radius 4cm and height 10cm.
1
Volume = πr²h = π × 4² × 10
2
Volume = π × 16 × 10 = 160π
Answer160π cm³ (≈ 502.7cm³)
A cylinder's radius r and height h are both perpendicular measurements.
Cones and spheres
The most common cone mistake is confusing the perpendicular height (h) with the slant height (l, the length of the sloped side). Volume uses h; the curved surface area uses l.
Volume of a cone = ⅓πr²h
Curved surface area of a cone = πrl
Volume of a sphere = (4/3)πr³
Surface area of a sphere = 4πr²
h (dashed, perpendicular) is used for volume. l (the sloped side) is used for the curved surface area — they are different lengths.
Worked Example 4 — Cone volume
Find the volume of a cone with radius 3cm and perpendicular height 10cm.
1
Volume = ⅓πr²h = ⅓ × π × 3² × 10
2
Volume = ⅓ × π × 9 × 10 = 30π
Answer30π cm³ (≈ 94.2cm³)
Compound 3D shapes
As with 2D compound shapes, split a 3D compound solid into parts you already know how to work with, then combine the results.
Worked Example 5 — Cone and hemisphere
A solid is made from a cone of radius 3cm and height 10cm, joined to a hemisphere of radius 3cm on its base. Find the total volume.
1
Cone volume = ⅓πr²h = ⅓ × π × 9 × 10 = 30π
2
Hemisphere volume = ½ × (4/3)πr³ = ½ × (4/3) × π × 27 = 18π
3
Total volume = 30π + 18π = 48π
Answer48π cm³ (≈ 150.8cm³)
Practice questions
Work through each question before checking the answers. Where a question describes a prism, cone, compound solid, or similar solids, the diagram shown is required to answer it. Use π = 3.14 or the π button on your calculator, and round to 1 decimal place unless told otherwise.
Foundation (Grade 3–5)
Q1A cuboid measures 6cm by 4cm by 3cm. Find its volume.Foundation
Q2A cuboid measures 6cm by 4cm by 3cm. Find its surface area.Foundation
Q3Find the volume of a cylinder with radius 4cm and height 10cm.Foundation
Q4The diagram shows a triangular prism. Find its volume.Foundation
Diagram for Question 4 (not to scale).
Q5Find the volume of a sphere with radius 3cm.Foundation
Higher (Grade 5–7)
Q6Find the surface area of a cylinder with radius 5cm and height 12cm.Higher
Q7The diagram shows a cone. Find its volume.Higher
Diagram for Question 7 (not to scale).
Q8Find the curved surface area of a cone with radius 5cm and slant height 13cm.Higher
Q9A cone has radius 3cm and perpendicular height 4cm. Find its curved surface area.Higher
Q10Find the surface area of a sphere with radius 7cm.Higher
Higher — Hard (Grade 8–9)
Q11The diagram shows a solid made from a cone and a hemisphere. Find the total volume.Grade 8–9
Diagram for Question 11 (not to scale). The label 9cm is the cone's height; 4cm is the radius (shared by the cone and hemisphere).
Q12The diagram shows a cylinder with a cone-shaped hole drilled from the top, all the way through. Find the remaining volume.Grade 8–9
Diagram for Question 12 (not to scale). r is the shared radius of the cylinder and the drilled cone; h is the shared height. The dashed lines show the cone-shaped hole.
Q13A sphere has a volume of 288π cm³. Find its radius.Grade 8–9
Q14A cylinder has a volume of 250π cm³ and a height of 10cm. Find its radius.Grade 8–9
Q15The diagram shows two similar cones. Their heights are in the ratio 2:3. The volume of the smaller cone is 32cm³. Find the volume of the larger cone.Grade 8–9
Diagram for Question 15 (not to scale). The dashed lines show each cone's perpendicular height. The two cones are mathematically similar.
Answers
Foundation (Q1–Q5)
Q172cm³(6 × 4 × 3)
Q2108cm²(2 × (24+18+12))
Q3502.7cm³(π × 16 × 10)
Q4120cm³(15 × 8)
Q5113.1cm³((4/3)π × 27)
Higher (Q6–Q10)
Q6534.1cm²(2π×25 + 2π×5×12)
Q7301.6cm³(⅓π × 36 × 8)
Q8282.7cm²(π × 5 × 13)
Q975.4cm²(slant height = 5 via Pythagoras, then π × 3 × 5)
Q10615.8cm²(4π × 49)
Higher — Hard (Q11–Q15)
Q11284.9cm³(cone 48π + hemisphere 128π/3)
Q12628.3cm³(cylinder 300π − cone 100π = 200π)
Q136cm(r³ = 216)
Q145cm(r² = 25)
Q15108cm³(scale factor (3/2)³ = 27/8, so 32 × 27/8)
Common mistakes
Common Mistake 1
Confusing height with slant height in cones
The perpendicular height (h) is used for volume. The slant height (l), the length of the sloped side, is used for the curved surface area. They are different lengths and are not interchangeable.
Common Mistake 2
Wrong units for volume and surface area
Volume is measured in cubic units (cm³), surface area in square units (cm²). Writing the wrong unit — or forgetting units altogether — loses marks even with the correct number.
Common Mistake 3
Using the diameter instead of the radius
Every volume and surface area formula on this page uses the radius. If a question gives a diameter, halve it before substituting.
Common Mistake 4
Adding a face that no longer exists in a compound solid
Where two solids join (like the flat circle between a cone and a hemisphere), that face is internal and shouldn't be included when finding the total surface area of the compound shape — though it's irrelevant for volume, which is simply added.
Exam tips
💡 Exam Tip 1
Sketch the solid and label r, h and l
Before substituting into any formula, sketch the shape and clearly mark which measurement is which — this prevents the height/slant height mix-up.
💡 Exam Tip 2
Write the formula before substituting numbers
Writing "Volume = ⅓πr²h" before putting numbers in secures method marks even if the final calculation goes wrong.
💡 Exam Tip 3
Split compound solids into simple 3D shapes
Just like compound 2D shapes, identify the simple solids that make up a compound one, calculate each separately, then add or subtract.
💡 Exam Tip 4
Keep answers in terms of π until the final step
Carrying π through your working (e.g. 30π rather than 94.2) avoids rounding errors compounding in multi-step problems.
Want to improve your grade faster?
If volume and surface area are still causing problems, Alamin's diagnostic approach identifies exactly which skills are missing and builds a targeted plan to address them — with AI-powered practice between sessions.