Grade 11 Measurement

Grade 11 measurement: surface area and volume of pyramids, cones, spheres and composite objects, and the effect of scaling, with worked CAPS examples.

Grade 11 measurement revises the surface area and volume of prisms, cylinders, pyramids, cones and spheres, and applies them to composite objects such as a cone on a hemisphere or a cylinder with a hollow inside. It is examined with Euclidean geometry in Paper 2.

The formulas are given, so the marks are for choosing the right ones, working out missing lengths (often with Pythagoras), and keeping track of which faces are included. You also use scale factors: multiplying every length by k multiplies area by k² and volume by k³.

Key concepts

  • Volume of a pyramid or cone = ⅓ × area of base × perpendicular height
  • Sphere: V = (4/3)πr³ and surface area = 4πr²
  • Cone: curved surface area = πrs, where s is the slant height
  • Composite objects: add or subtract volumes; include only exposed faces in surface area
  • Using Pythagoras to find slant or perpendicular heights
  • Scaling: area × k², volume × k³

Worked examples

Find the volume of a cone with radius 3 cm and height 4 cm. Leave π in your answer.

V = ⅓πr²h = ⅓ × π × 9 × 4 = 12π cm³.

Find the total surface area of the same cone.

Slant height s = √(3² + 4²) = 5 cm. Total surface area = πr² + πrs = 9π + 15π = 24π cm².

A closed cylinder has radius 5 cm and height 10 cm. Find its total surface area. Leave π in your answer.

Total surface area = 2πr² + 2πrh = 50π + 100π = 150π cm².

A solid is a hemisphere of radius 3 cm with a cone of height 4 cm on top. Find its volume.

Hemisphere: ½ × (4/3)π(3)³ = 18π. Cone: ⅓π(3)²(4) = 12π. Total = 30π ≈ 94.25 cm³.

Common mistakes

  • Using the slant height instead of the perpendicular height for volume.
  • Counting the joining face in the surface area of a composite object.
  • Forgetting to halve the sphere formulas for a hemisphere.
  • Mixing units such as mm and cm.

Exam tips

  • Sketch the solid and label every length before substituting.
  • Write the formula first: it earns a method mark even if the arithmetic slips.
  • Leave π in the answer unless the question asks for a decimal.

Frequently asked questions

Are the measurement formulas given in the Grade 11 exam?

Yes, the formulas for surface area and volume are normally provided. You need to know which one to use and how to find missing lengths.

What is the slant height of a cone?

The distance from the apex to the edge of the base, along the surface. Find it with Pythagoras: s² = r² + h².

How do I find the surface area of a composite object?

Add the areas of the exposed surfaces only. Faces where two solids are joined are inside the object, so leave them out.

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