How many degrees are there in a triangle?

The three angles of any triangle add up to 180 degrees.

What are vertically opposite angles?

When two straight lines cross, the angles opposite each other are equal. They are called vertically opposite angles.

What is the difference between radius and diameter?

The radius goes from the centre to the edge of the circle. The diameter goes right across through the centre, so it is twice as long.

Year 6 angles, circles and 3D shapes lesson to download as PDF

This Year 6 angles, circles and 3D shapes lesson is written for pupils aged 10-11 and follows the National Curriculum. In short, it shows you how to find missing angles in triangles and quadrilaterals, name parts of a circle and draw nets of 3-D shapes. Before you start, it helps to revise algebra: formulae, sequences and missing numbers and converting units, area, perimeter and volume, because this lesson builds on them.

Table of Contents

Year 6 angles, circles and 3D shapes lesson: what you will learn

Angle facts help us find missing angles. We also learn the parts of a circle and the nets of 3D shapes.

Starter activity

A builder is fixing a triangular frame for a roof. She needs the angle at the top and the two angles at the bottom to be exactly right. She also needs a circular window and a box with a lid that folds out of flat card. In this chapter you will learn the angle facts that help to find missing angles, the parts of a circle and how 3D shapes are made from nets.

Angle facts you need to know

Rule: angle facts

Angles on a straight line add up to 180°. Angles at a point add up to 360°. Angles in a triangle add up to 180°. Angles in a quadrilateral add up to 360°. Vertically opposite angles are equal.

Definition: isosceles and equilateral triangles

An isosceles triangle has two equal sides and two equal angles. An equilateral triangle has three equal sides and three angles of 60°.

Finding missing angles

Method: finding a missing angle

Decide which angle fact applies.
Add up the angles you know.
Subtract from the total (180° or 360°).

Example 1

A triangle has angles of 50° and 60°. What is the third angle?

50 + 60 = 110. 180 − 110 = 70°.

Example 2

An isosceles triangle has an angle of 40° at the top. Find the base angles.

The base angles together are 180 − 40 = 140°. They are equal, so each is 140 ÷ 2 = 70°.

Example 3

Three angles of a quadrilateral are 90°, 85° and 110°. Find the fourth.

90 + 85 + 110 = 285. 360 − 285 = 75°.

Example 4

Two straight lines cross and one angle is 48°. Find the other three angles.

The angle opposite is also 48° (vertically opposite). The other two lie on a straight line with it, so each is 180 − 48 = 132°.

Watch out

Do not mix up 180° (triangle, straight line) and 360° (quadrilateral, point). Diagrams are often not drawn accurately, so calculate rather than measure.

Drawing 2D shapes accurately

To draw a triangle from two angles and a side, draw the side first and use a protractor at each end. Use a ruler for lines and measure to the nearest millimetre.

Parts of a circle

Definition: parts of a circle

The centre is the middle point. The radius goes from the centre to the edge. The diameter goes across the circle through the centre. The circumference is the distance round the edge.

Key fact: radius and diameter

The diameter is twice the radius, so \(d=2r\). A circle with radius 4 cm has diameter 8 cm.

Example 5

A circle has diameter 14 cm. What is its radius?

14 ÷ 2 = 7 cm.

3D shapes and nets

Definition: net

A net is a flat shape that folds up to make a 3D shape. A cube has 6 faces, 12 edges and 8 vertices.

Key fact: common 3D shapes

A cuboid has 6 faces, a triangular prism has 5 faces, and a square-based pyramid has 5 faces and 8 edges.

Example 6

How many square faces are in the net of a cube?

A cube has 6 faces and all are squares, so its net has 6 squares.

More about angles and shapes

Method: finding missing angles in a diagram

Look for a straight line (180°), a point (360°), a triangle (180°) or a quadrilateral (360°).
Write down the angle fact you are using.
Add the angles you already know.
Subtract that total from 180° or 360° to find the missing angle.

Example 7

An isosceles triangle has one angle of 80°, and it is the angle between the two equal sides. Find the other angles.

The other two angles are equal, because it is an isosceles triangle. Angles in a triangle add up to 180°, so the other two angles add up to 180 − 80 = 100°. Each one is 100 ÷ 2 = 50°.

Example 8

Find the missing angle at a point: 120°, 95° and 80° with a fourth angle \(x\).

Angles at a point add up to 360°. 120 + 95 + 80 = 295, so \(x=360-295=\) 65°.

Example 9

A circle has a diameter of 11 cm. Find the radius and set the compasses to draw it.

The radius is 11 ÷ 2 = 5.5 cm. Open the compasses until the point and the pencil are 5.5 cm apart, using a ruler. The pencil then draws a circle with a diameter of 11 cm.

Example 10

Check that a triangular prism fits the rule vertices + faces − edges = 2.

A triangular prism has 6 vertices, 5 faces and 9 edges. 6 + 5 − 9 = 2. The rule works for all solid shapes with flat faces.

Watch out: common mistakes

Diagrams in a test are often not drawn accurately, so never measure an angle that is labelled “not drawn accurately”: calculate it. Do not use 180° when the shape is a quadrilateral, and do not use 360° for a triangle. Also, the radius is half the diameter, so a circle that is 10 cm wide needs the compasses set to 5 cm.

Reading and using a protractor

Put the centre of the protractor on the point (the vertex) of the angle, and line up the baseline with one arm. Read the scale that starts at 0° on that arm. If the angle is acute (less than 90°), the answer should be less than 90°, so you can check that you have chosen the right scale.

Did you know?

The number 360 appears in the circle because the ancient Babylonians counted in groups of 60. They also thought that the Sun took about 360 days to go round the sky, so they split a circle into 360 equal parts. We still use degrees today, 4 000 years later.

Otto’s tip

Write the angle fact next to every step, such as “angles in a triangle add up to 180°”. It helps you spot mistakes!

How to use this Year 6 angles, circles and 3D shapes lesson

First, read the key points so that you know where the Year 6 angles, circles and 3D shapes lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the angles, circles and 3D shapes worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the angles, circles and 3D shapes test and mark it with the test answers.

Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 6 angles, circles and 3D shapes lesson gives you everything you need for the chapter.

To sum up, regular practice works better than one long session. For instance, ten minutes a day is often enough. Besides, mistakes are useful, because they show what to revise next. Then, once a method feels easy, try a harder question. As a result, you build confidence step by step.