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Year 6 maths · Key Stage 2 · Geometry – properties of shapes

Year 6 geometry: missing angles, circles and 3-D shape nets

Year 6 geometry combines accurate drawing with reasoning. Children draw 2-D shapes from given lengths and angles, build 3-D shapes and make their nets, name the parts of a circle, classify shapes by their properties, and find missing angles in triangles, quadrilaterals and regular polygons. This year the emphasis moves further from measuring angles towards working them out.

Angles, circles and nets, explained

Children draw shapes using a ruler and a protractor, following instructions such as “draw a triangle with sides of 6 cm and 4 cm and an angle of 50° between them”. They use the conventional markings: a small square for a right angle, ticks to show equal sides, arrows for parallel lines, and capital letters for corners. Accurate drawing is also a check on reasoning: if the calculated angle and the measured angle differ a lot, one of them is wrong.

A net is a 3-D shape opened out flat. A cube’s net is six squares joined edge to edge (there are 11 different arrangements that fold up into a cube). A triangular prism needs two triangles and three rectangles, and a square-based pyramid needs one square and four triangles. Children build shapes from card, describe them using faces, edges and vertices (a cube has 6, 12 and 8) and imagine how a net would fold.

Children compare and classify shapes by their properties, such as side lengths, parallel sides and angles, for example seeing that a square is a special kind of rectangle and also a special kind of rhombus. They use two key facts to find unknown angles. The angles inside any triangle add up to 180°, and the angles in any quadrilateral add up to 360°. A regular polygon has equal sides and equal angles, so a regular hexagon’s six angles are all the same. Some schools write these facts as formulae, for example a = 180 − (b + c), but the reasoning is the point.

Angles also follow simple rules where lines meet. Angles around a point add up to 360°, angles on a straight line add up to 180°, and when two straight lines cross, the angles facing each other (vertically opposite angles) are equal. Children use these in short chains of reasoning: “this angle is 115° because it is on a straight line with 65°, and the angle opposite it is also 115°”. Explaining each step in words counts as much as the number.

A circle has three named parts that children must be able to label. The radius runs from the centre to the edge, the diameter goes right across through the centre, and the circumference is the distance all the way round. The diameter is twice the radius (d = 2 × r). Measuring round a tin with string shows that the circumference is a little more than three times the diameter. Calculating the circumference with the number called pi (π) comes later, in Key Stage 3.

Words your child will hear

Net
A flat pattern of faces that folds up to make a 3-D shape.
Vertex
A corner of a shape. The plural is vertices.
Regular polygon
A polygon with all sides equal and all angles equal, such as a square or a regular hexagon.
Vertically opposite angles
The pairs of equal angles facing each other when two straight lines cross.
Radius
The distance from the centre of a circle to its edge.
Diameter
The distance across a circle through its centre; twice the radius.
Circumference
The distance all the way round a circle.

What the National Curriculum says

In Year 6, pupils should be taught to:

  • Draw 2-D shapes using given dimensions and angles
  • Recognise, describe and build simple 3-D shapes, including making nets
  • Compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
  • Illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius
  • Recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

What is the size of each angle inside a regular hexagon?

  1. Mark the centre of the hexagon and join it to every corner. This splits the hexagon into 6 identical triangles.
  2. The angles at the centre make a full turn, 360°. There are 6 equal angles, so each is 360° ÷ 6 = 60°.
    360 ÷ 6 = 60
  3. Each triangle has two sides that are the same length (both go from the centre to a corner), so it is isosceles and its other two angles are equal. The three angles add up to 180°, so the other two share 180° − 60° = 120°. Each is 120° ÷ 2 = 60°.
    180 − 60 = 120
120 ÷ 2 = 60
  4. At each corner of the hexagon, two of these 60° angles sit side by side, so the angle inside the hexagon is 60° + 60° = 120°.
  5. Check another way: split the hexagon into 4 triangles by drawing lines from one corner. 4 × 180° = 720°, and 720° ÷ 6 = 120°. Both methods agree.

Each angle inside a regular hexagon is 120°.

Common mistakes and how to help

Reading the wrong scale on the protractor

Protractors have two scales. A child lines up the zero correctly but reads the other row, so an angle of 55° is written as 125°.

What helps: Decide first whether the angle is smaller or larger than a right angle. If it is clearly acute, the answer must be under 90°. Always start from the zero that lines up with one arm.

Thinking an angle is bigger if its arms are longer

The drawn lines of a bigger-looking angle may be longer, and the child compares the picture’s size, not the amount of turn.

What helps: Show two angles that are the same size with lines of different length, or draw one angle and extend its arms. Describe an angle as a turn, like opening a door.

Mixing up radius and diameter

Both words describe lines in a circle, and in a diagram only one line may be shown. A child told the diameter is 12 cm says the radius is 24 cm.

What helps: The radius reaches from the centre to the edge, so it is the shorter one. The diameter is “double the radius”. Drawing both on the same circle and measuring them fixes the link.

Assuming any six squares joined together fold into a cube

The pieces are the right number and the right shape, so the arrangement looks plausible.

What helps: Cut six paper squares and try. A 2 by 3 block can’t fold up because the faces overlap, while a cross shape, with a row of four and one square above and below, will. Folding it for real beats arguing about it.

How to help at home

  • Tear and fit the corners. Draw any triangle on paper, tear off its three corners and place them side by side. They make a straight line, which shows the angles add up to 180°. Do the same with a quadrilateral and the corners fill a full turn, 360°.
  • Take a box apart. Open out an empty cereal or tissue box along its glued edges (an adult does any cutting). Your child sees a net, counts the faces and works out how it folds. Then draw a different net for the same box.
  • Circle measuring. Choose a tin or a plate. Measure across the widest part for the diameter, halve it for the radius, and use string for the circumference. It is a little over three times the diameter whichever circle you choose.
  • Angle hunt. Look for angles around the house: a door opening, the corner of a table, the hands of a clock. Estimate each as being smaller than, equal to or bigger than a right angle, then check with a protractor if you have one.
  • Draw it accurately. Give your child instructions to follow with a ruler and protractor, such as a triangle with sides of 5 cm and 7 cm and a 40° angle between them. Compare the drawing with someone else’s following the same instructions.

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