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Surface Area of a Cone / Sphere

FoundationHigherAQAEdexcelOCR

Surface Area of a Cone / Sphere is a key geometry topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on surface area of cones and spheres, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Curved surface area of a sphere = 4πr²; of a cone = πrl.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

The surface areas of cones and spheres have their own formulae, both provided on the formula sheet.

A cone's curved surface area is \(\pi r l\), where \(l\) is the slant height. Adding the circular base gives a total of \(\pi r l + \pi r^2\). A sphere's surface area is \(4\pi r^2\).

The slant height is the distance from the apex down the sloping side to the edge of the base, not the vertical height. If a question gives the vertical height instead, Pythagoras' theorem finds the slant height, since the radius, height and slant form a right-angled triangle.

Revision notes

Cones

Curved surface area is \(\pi r l\) with the slant height. The total surface area adds the base, giving \(\pi r l + \pi r^2\).

For radius \(3\)cm and slant height \(5\)cm: curved area is \(15\pi\)cm², total is \(15\pi + 9\pi = 24\pi\)cm².

Finding the slant height

The radius, vertical height and slant height form a right-angled triangle, so \(l = \sqrt{r^2 + h^2}\).

With radius \(3\)cm and height \(4\)cm, the slant height is \(\sqrt{9 + 16} = 5\)cm.

Spheres

A sphere's surface area is \(4\pi r^2\), which is exactly four times the area of its great circle.

A hemisphere's curved surface is half of this, but its total surface area also includes the flat circular face.

Key points

  • Cone curved surface area is \(\pi r l\).
  • \(l\) is the slant height, not the vertical height.
  • Total cone surface area adds \(\pi r^2\) for the base.
  • Find the slant height with Pythagoras.
  • Sphere surface area is \(4\pi r^2\).
  • Surface area uses squared units.

Worked examples

Example 1

Find the curved surface area of a cone with radius \(5\)cm and slant height \(12\)cm, in terms of \(\pi\).

Working

\[\pi r l = \pi \times 5 \times 12\]substitute into the formula
\[= 60\pi\text{cm}^2\]state the curved surface area

Example 2

Find the surface area of a sphere with radius \(6\)cm, in terms of \(\pi\).

Working

\[4\pi r^2 = 4\pi \times 36\]square the radius first
\[= 144\pi\text{cm}^2\]state the surface area

Example 3

A cone has radius \(6\)cm and vertical height \(8\)cm. Find its slant height.

Working

\[l = \sqrt{6^2 + 8^2}\]the radius, height and slant form a right-angled triangle
\[= \sqrt{100}\]add the squares
\[= 10\text{cm}\]take the square root

Common mistakes

  • Using the vertical height in \(\pi r l\).

    The formula needs the slant height. Use Pythagoras if only the vertical height is given.

  • Forgetting the base of a cone.

    Total surface area includes the circular base as well as the curved surface.

  • Using cubed units.

    Surface area is an area, so cm².

  • Halving a sphere's surface area for a hemisphere without adding the flat face.

    The total also includes the circular face.

Exam tips

  • Check whether the question gives slant or vertical height.
  • Use Pythagoras to convert between them.
  • Read whether curved or total surface area is wanted.
  • Always use squared units.

Key terms

Slant height
The distance from apex to base edge along the surface.
Curved surface area
The area of the sloping surface, excluding the base.
Hemisphere
Half a sphere.
Great circle
The largest circle that fits on a sphere.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.