Grade 10 Measurement: Surface Area and Volume

Grade 10 measurement: surface area and volume of prisms, cylinders, pyramids, cones and spheres, and the effect of scaling dimensions, with worked examples.

Grade 10 measurement extends area and volume to right prisms, cylinders, pyramids, cones and spheres, and to objects made from combinations of them. It is examined with Euclidean geometry in Paper 2, and the formulas are usually given, so the marks are for choosing the right formula and substituting correctly.

You also study what happens when dimensions are multiplied by a factor k: lengths scale by k, areas by k² and volumes by k³. That idea appears again whenever a question asks how doubling or halving a measurement changes the answer.

Key concepts

  • Volume of a right prism or cylinder = area of base × height
  • Volume of a pyramid or cone = ⅓ × area of base × height
  • Volume of a sphere = (4/3)πr³ and surface area = 4πr²
  • Surface area as the sum of the areas of all faces (nets)
  • Scaling by k multiplies area by k² and volume by k³

Worked examples

Find the surface area of a cube with edges of 4 cm.

A cube has 6 square faces: 6 × 4² = 96 cm². Its volume is 4³ = 64 cm³.

Find the volume of a cone with radius 3 cm and height 7 cm (π ≈ 3.142).

V = ⅓πr²h = ⅓ × 3.142 × 9 × 7 ≈ 65.98 cm³.

The radius of a sphere is doubled. By what factor does its volume increase?

V = (4/3)πr³, so replacing r with 2r multiplies the volume by 2³ = 8.

Find the volume and total surface area of a closed cylinder with radius 5 cm and height 10 cm. Leave π in your answers.

Volume = πr²h = π × 25 × 10 = 250π cm³ (about 785.4 cm³). Total surface area = 2πr² + 2πrh = 50π + 100π = 150π cm².

Common mistakes

  • Using slant height instead of perpendicular height in a volume formula.
  • Forgetting the ⅓ for pyramids and cones.
  • Mixing units, e.g. a radius in mm and a height in cm.
  • Counting only the curved surface of a closed cylinder.

Exam tips

  • Write the formula first, then substitute; method marks are given for the formula.
  • Convert all lengths to the same unit before calculating.
  • Give units: cm² for area, cm³ for volume.

Frequently asked questions

Are measurement formulas given in the Grade 10 exam?

Usually yes: the formulas for the surface area and volume of the solids are provided in the question or on an information sheet. You need to know when to use each one.

What is the difference between surface area and volume?

Surface area is the total area of the outside faces (square units). Volume is the space inside (cubic units).

What happens to the volume if every length is tripled?

Volume scales by the cube of the factor, so it is multiplied by 3³ = 27.

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