Year 10: Three-dimensional shapes and pyramids

In Year 10 Geometry, you are expected to calculate properties of 3D shapes such as prisms, cylinders, and pyramids. The main focus is understanding faces, edges, and vertices, and then finding surface area accurately.

Surface area means the total area of all outer faces. A net is often the easiest way to organise the calculation.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

Reading 3D shapes

Start by identifying the base shape. A prism has the same cross-section all the way through, while a pyramid has triangular faces meeting at an apex.

Skill quiz: Identify the solid and its faces

Practise recognising the base, cross-section and faces used in three-dimensional measurements.

Calculate the properties of basic three-dimensional shapes

Surface area of a prism

Example 1: Find the surface area of a rectangular prism with dimensions

4 cm

3 cm

, and

2 cm

There are three pairs of equal faces.
Area $= 2 (4 \times 3) + 2 (4 \times 2) + 2 (3 \times 2)$ .
$= 24 + 16 + 12 = 52 {cm}^{2}$ .

Skill quiz: Include every prism face

List all faces and add their areas, using the same units throughout.

Calculate the properties of basic three-dimensional shapes

Surface area of a pyramid

For a square pyramid, add the area of the square base and the areas of the four triangular faces.

Example 2: A square pyramid has base side length

6 cm

and slant height

5 cm

. Find the surface area.

Base area $= 6 \times 6 = 36 {cm}^{2}$ .
One triangular face area $= 1 / 2 \times 6 \times 5 = 15 {cm}^{2}$ .
Four triangular faces give $4 \times 15 = 60 {cm}^{2}$ .
Total surface area $= 36 + 60 = 96 {cm}^{2}$ .

Skill quizzes: Add the base and triangular faces

Start with Level 1 pyramid surface areas, then progress to Level 2 and check the slant height.

Why nets help

A net turns the solid into flat shapes you already know how to measure. This is especially helpful in pyramid questions where students often forget one or more triangular faces.

Example 3: A triangular prism has two triangular ends and three rectangular side faces. A net makes it obvious that all five faces must be included in the final total.

Skill quizzes: Use a net to count faces

Sketch a net while solving the surface-area questions so that every face is included.

Common mistakes

Confusing slant height with vertical height in pyramid questions.
Leaving out one face when adding surface areas.
Mixing units, such as centimetres and metres, in the same calculation.

Skill quiz: Check heights, faces and units

Revisit the harder pyramid questions and distinguish vertical height from the height of a triangular face.

Level 2: Calculate the surface area of a pyramidPremium