Measurement: Surface Area, Volume & Scaling (Grade 10)

Surface area and volume of prisms, cylinders, cones, spheres and pyramids, and what happens to area and volume when every dimension is multiplied by a factor k.

Grade 10 adds the cone, sphere and pyramid to the prisms and cylinders you already know — and asks one new question: what happens when you scale a solid up?

1The formulae

The three curved solids added in Grade 10. Note the ⅓ in the cone — it holds a third of the cylinder that surrounds it.

Prism: V = area of cross-section × height
Cylinder: V = πr²h · SA = 2πr² + 2πrh
Cone: V = ⅓πr²h · SA = πr² + πrℓ
Sphere: V = 43πr³ · SA = 4πr²
Pyramid: V = ⅓ × base area × height

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The cone and pyramid both carry a ⅓; the cylinder and prism do not. That single detail is the most common lost mark in this topic.

Worked ExampleA cylinder has r = 5 cm and h = 12 cm. Find its volume (π ≈ 3,14).

1
V = πr²h = 3,14 × 25 × 12.
Square the radius first.
2
V = 942 cm³.

Worked ExampleA cone has r = 3 cm and h = 4 cm. Find its volume.

1
V = ⅓πr²h = ⅓ × 3,14 × 9 × 4.
2
V = ⅓ × 113,04 = 37,68 cm³.
Exactly a third of the matching cylinder.

Worked ExampleA sphere has r = 6 cm. Find its surface area.

1
SA = 4πr² = 4 × 3,14 × 36.
2
SA = 452,16 cm².

2The effect of scaling

If every dimension of a solid is multiplied by k:

lengths × k · surface area × k² · volume × k³

Worked ExampleEvery dimension of a box is doubled. What happens to its volume?

1
k = 2, and volume scales by k³.
Volume involves three dimensions.
2
2³ = 8, so the volume becomes 8 times bigger.
Its surface area becomes 2² = 4 times bigger.

⚠️

Doubling the dimensions does not double the volume — it multiplies it by 8. Area goes up by k² and volume by k³, never by k.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
The diagram below shows a cylinder. Calculate its volume. (π ≈ 3,14)
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3,14 × 25 × 12 = 942 cm³
2
The diagram below shows a cone. Calculate its volume. (π ≈ 3,14)
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⅓ × 3,14 × 9 × 4 = 37,68 cm³
3
Calculate the volume of the sphere in the diagram below. (π ≈ 3,14)
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43 × 3,14 × 27 = 113,04 cm³
4
Calculate the surface area of the sphere in the diagram below. (π ≈ 3,14)
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4 × 3,14 × 36 = 452,16 cm²
5
The diagram below shows a pyramid. Calculate its volume.
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⅓ × 30 × 9 = 90 cm³
6
A solid's dimensions are all doubled. What happens to its volume?
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It becomes 8 times bigger (2³).
7
A solid's dimensions are all tripled. What happens to its surface area?
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It becomes 9 times bigger (3²).
8
The diagram below shows a triangular prism. Calculate its volume.
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½ × 6 × 4 = 12 cm²; 12 × 10 = 120 cm³
9
Calculate the volume of the cylinder in the diagram below. (π ≈ 3,14)
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3,14 × 16 × 8 = 401,92 cm³
10
Which two solids have a ⅓ in their volume formula?
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The cone and the pyramid