Key Stage 3 maths · Geometry and measures

KS3 constructions and scale drawings: a guide for parents

Accurate drawing uses a ruler, a protractor and a pair of compasses, and the standard constructions use only a ruler and compasses, so a result is shown to be true rather than measured by eye. In Years 7 to 9 pupils draw and measure lines and angles, interpret scale drawings, carry out the standard constructions, use correct geometric notation and learn when two triangles must be identical.

Constructions and scale drawings, explained

Accuracy is the first skill. Pupils draw lines to the nearest millimetre and angles to the nearest degree, and they learn the routines that avoid common slips, such as lining the protractor’s centre point on the vertex and reading the correct scale. They also measure angles in diagrams, including reflex angles (more than 180°). A scale drawing is a smaller or larger version of reality in which every length is multiplied by the same number. A scale of 1 cm to 2 m means 8 cm on paper stands for 16 m in real life, and the same scale lets you read a length off the paper and convert it back.

There are three standard constructions that pupils learn to derive and use, and the aim is to understand why each works. To make the perpendicular bisector of a line segment, you draw arcs of equal radius from each end so they cross above and below the line, then join the crossing points. The result cuts the segment exactly in half at a right angle. To bisect an angle, you draw an arc from the vertex across both arms, then equal arcs from where it crosses, and join the vertex to where those arcs meet. To draw a perpendicular from a point to a line, you draw an arc centred on the point that cuts the line twice, then construct the perpendicular bisector of those two marks. A perpendicular at a point on a line works the same way, starting from two marks the same distance either side of the point. Constructions are done with compasses, not by measuring, and the arcs must be left visible as evidence.

Pupils also learn that the shortest distance from a point to a line is the perpendicular distance, the line from the point that meets the line at a right angle. It sounds obvious but it is the idea behind finding the nearest road, the shortest ladder to a wall or the radius of a circle that just touches a line.

Geometry has its own language and notation. Points are named with capital letters, a line segment between A and B is written AB, and parallel lines are marked with matching arrows. A right angle is marked with a small square, and equal sides or angles with matching ticks or arcs. In triangle ABC the angles are written with the vertex in the middle, so angle ABC is the angle at B, and the side opposite vertex A is labelled with a lowercase a. Pupils describe, sketch and draw regular polygons and other polygons with reflective or rotational symmetry using these conventions.

Finally, pupils learn the conditions that make two triangles congruent, meaning identical in shape and size. Three sides the same (SSS), two sides and the angle between them (SAS), two angles and a corresponding side (ASA or AAS), and a right angle with the hypotenuse and one other side (RHS) each guarantee congruence. Pupils also see why three equal angles do not: a small and a large triangle can have the same angles.

Words your child will hear

Perpendicular bisector: A line that cuts a segment in half and meets it at a right angle.
Arc: Part of a circle’s edge, drawn with compasses when making a construction.
Bisect: To cut exactly in half, such as bisecting an angle or a line.
Scale: The ratio between a length in a drawing and the real length, such as 1 cm to 5 m.
Vertex: A corner where two lines meet. The angle ABC has its vertex at B.
Congruent: Identical in shape and size. One shape can be turned, flipped or slid on top of the other.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Draw and measure line segments and angles in geometric figures, including interpreting scale drawings
Derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line
Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric
Use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles

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

Try it: finding a point the same distance from two trees

Two trees stand at A and B, 8 m apart. Use a scale of 1 cm to 1 m to find where a bench could stand so it is exactly the same distance from both trees. Which point on that line is closest to the trees?

Draw the line AB 8 cm long with a ruler and mark the ends clearly. At this scale 1 cm stands for 1 m, so 8 cm is the 8 m between the trees.
Set the compasses to a radius of about 5 cm, comfortably more than half of AB (4 cm). If the radius is too small the arcs will not cross.
Place the compass point on A and draw one arc above the line and one below. Without changing the radius, repeat from B so that the arcs cross at two points.
Use a ruler to join the two crossing points. Leave the arcs in place. This line is the perpendicular bisector of AB. Every point on it is the same distance from A as from B.
Check by measuring: the line meets AB at its midpoint, 4 cm from each end, and a protractor shows a 90° angle. That midpoint is the closest point on the bisector to the trees, and in real life it is 4 m from each.

The bench can go anywhere on the perpendicular bisector of AB. The closest point is the midpoint of AB, 4 m from each tree.

Common mistakes and how to help

Measuring instead of constructing

A ruler and protractor feel quicker and more reliable than compasses, so pupils find the middle of a line by measuring and rubbing out the evidence.

What helps: Explain that a construction proves the result without measuring. Ask your child to do both ways and compare, then say why the arcs must stay on the page.

Changing the compass width between arcs

The compasses slip, or the pupil sets a new width for the second arc. The arcs then cross at points that are not equal distances from both ends.

What helps: Tighten the compasses, use a pencil that is sharp, and keep the width fixed for each pair of arcs. A slip makes the final line visibly off centre, which is a good check.

Thinking three equal angles make triangles congruent

Two triangles with the same angles look similar, so it seems natural they are the same. But one can be a scaled-up copy of the other.

What helps: Draw a 3-4-5 triangle and one twice as big on squared paper. Their angles match but the sides do not. Congruence needs at least one side length.

Misreading the protractor scale

Most protractors have two sets of numbers. A pupil starts on the wrong scale and reports 110° for an angle that is really 70°.

What helps: Estimate first: is the angle acute or obtuse? An acute angle cannot be 110°. Always start counting from 0 on the baseline.

How to help at home

Draw a plan of your home. Choose a scale such as 1 cm to 50 cm, measure one room with a tape and draw it. Add furniture as scaled rectangles and see whether a new sofa would fit through the plan.
Use a map scale. Open a paper map or an atlas and measure a journey with a ruler. If 1 cm stands for 2 km, how far is it in real life? Check against the distance given.
Draw patterns with compasses. Draw a circle, then keep the compasses at the same radius and step the point round the edge, marking it six times. The marks land back at the start, and joining them gives a regular hexagon. Joining alternate points makes an equilateral triangle.
Fold to bisect. Fold a piece of paper so two ends of a line match and the crease is the perpendicular bisector. Folding an angle onto itself makes the angle bisector, a handy check of a construction.

Builds on

Year 6: Units, area and volume
Year 6: Angles, circles and nets
Year 6: Four-quadrant coordinates, translation, reflection

Leads on to

Key Stage 4: Enlargement, similarity and vectors
Key Stage 4: Circles, solids and bearings
Key Stage 4: Pythagoras and trigonometry
Key Stage 4: Circle theorems and vector proof
Key Stage 4: Sine rule and cosine rule