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Shape and geometry

Geometry is the part of maths about shape, size and position. In primary school it covers naming and describing 2D shapes (flat shapes like squares and triangles) and 3D shapes (solid shapes like cubes and spheres), symmetry, angles, and position and direction: describing where things are and how they move, including coordinates.

What children learn, year by year

This follows the National Curriculum in England and the EYFS framework for Reception. US grades are roughly one year behind the UK year number.

The big ideas

1. 2D shapes: sides and vertices

A 2D shape is flat. A shape with straight sides that join up all the way round is a polygon. Its corners are called vertices (one is a vertex). Polygons are named by how many sides they have:

A regular polygon has all its sides the same length and all its angles the same size. An irregular one doesn't. A stop sign is a regular octagon. Any 4-sided shape is a quadrilateral, and that family includes the square, rectangle, rhombus, parallelogram, trapezium and kite. A square is a special kind of rectangle, because it has four right angles. Triangles can be equilateral (all sides equal), isosceles (two sides equal), scalene (no sides equal) or right-angled.

2. 3D shapes: faces, edges and vertices

A 3D shape is solid. Its flat surfaces are faces, the lines where two faces meet are edges, and the corners are vertices.

Spheres, cylinders and cones have curved surfaces. A sphere has one curved surface; a cylinder has two flat circular faces and one curved surface.

3. Symmetry

A line of symmetry (or mirror line) splits a shape into two halves that are reflections of each other. If you cut the shape out and fold it along the line, the halves match exactly. A regular polygon has as many lines of symmetry as it has sides. A circle has an infinite number: any line through its centre works.

4. Angles are turns

An angle is an amount of turn. Children first meet turns with their bodies: a quarter turn, a half turn, a whole turn. A quarter turn is a right angle, the square corner you see on a book or a door frame. Two right angles make a half turn, and four make a whole turn.

In Year 5, children measure angles in degrees. A whole turn is 360°, a half turn (a straight line) is 180° and a right angle is 90°. Angles get their names from their size:

• Acute: less than 90°
• Right angle: exactly 90°
• Obtuse: more than 90° but less than 180°
• Reflex: more than 180° but less than 360°

To measure with a protractor: put the small cross at the centre of the protractor exactly on the vertex, line up the zero line with one arm, then follow the scale that starts at 0 on that arm round to the other arm. Protractors have two scales, and reading the wrong one is the most common slip: it gives 130° for a 50° angle. Ask "is this angle acute or obtuse?" first, so your child knows roughly what answer to expect.

5. Angle facts

From Year 5 and Year 6, children use a few facts to find missing angles without measuring:

• Angles on a straight line add up to 180°. If one angle is 35°, the other is 180° − 35° = 145°.
• Angles around a point add up to 360°.
• The angles inside a triangle add up to 180°. If two angles are 50° and 70°, the third is 180° − 120° = 60°.
• The angles inside a quadrilateral add up to 360°.
• Where two straight lines cross, the angles opposite each other (vertically opposite angles) are equal.

6. Position, direction and coordinates

Young children describe position with words like above, below, next to, left and right, and give directions such as "forward 3 steps, then a quarter turn clockwise". From Year 4, they use coordinates to name a point on a grid. The first number goes across (the x-axis) and the second goes up (the y-axis): to find (3, 5), go 3 across and 5 up. "Along the corridor, then up the stairs" helps children remember the order.

A translation slides a shape without turning it. Move the point (2, 1) 3 squares right and 2 squares up and it lands on (5, 3). A reflection flips a shape over a mirror line, and each point ends up the same distance from the line on the other side. In both cases the shape keeps its size. In Year 6, the grid extends to negative numbers, so reflecting the point (2, 3) in the y-axis gives (−2, 3).

Common mistakes to look out for

• "It's a diamond": a square turned on its corner is still a square. A diamond-like shape with equal sides but no right angles is a rhombus.
• A square isn't a rectangle: it is. It's a rectangle whose sides are all the same length.
• Only knowing the "usual" picture: recognising a triangle only when it sits flat on its base. Show shapes at all angles, sizes and colours.
• Diagonals as lines of symmetry: folding a rectangle along its diagonal doesn't make the halves match. Try it with paper.
• Longer arms, bigger angle: the size of an angle is the amount of turn, not the length of the lines.
• Coordinates the wrong way round: going up before across. Across first, then up.

Five-minute shape games for home

• Shape hunt. Find a cylinder, a cuboid and a sphere in the kitchen. Count the faces, edges and vertices of a cereal box together.
• Butterfly paint. Drop paint on one half of a sheet of paper, fold along the middle and open it. The fold is a line of symmetry.
• Robot directions. Your child gives you instructions to walk to the sofa: "forward 4 steps, quarter turn anticlockwise, forward 2."
• Coordinate battleships. Draw two small grids, hide some "ships" on points, and take turns calling coordinates to find each other's.

Free shape and geometry worksheets

Print these for practice at home or in class. Each one has an answer sheet on the second page.

2D shapesYear 1, Year 2 · Names, sides and verticesDownload PDF

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