Key Stage 3 maths · Geometry and measures

KS3 Pythagoras’ Theorem and trigonometry: a parent’s guide

Pythagoras’ Theorem links the three sides of a right-angled triangle, and it is one of the most useful results in school maths. In Years 7 to 9 pupils use angle facts, congruence and similarity to explain why results are true, write simple proofs, apply Pythagoras’ Theorem to find missing sides, and meet the trigonometric ratios that come from similar triangles.

Pythagoras and trigonometry, explained

In a right-angled triangle the longest side, opposite the right angle, is called the hypotenuse. Pythagoras’ Theorem says that the square of the hypotenuse equals the sum of the squares of the other two sides, written a² + b² = c². Geometrically, the squares drawn on the two shorter sides have a total area equal to the square drawn on the hypotenuse. Pupils test this on a 3, 4, 5 triangle (9 + 16 = 25) and then apply it to find a missing side. It also works in reverse: if three sides satisfy a² + b² = c², the triangle must contain a right angle, which is how builders check a corner.

The method depends on what is missing. To find the hypotenuse, square the two shorter sides, add and take the square root. To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root. Pupils are taught to sketch the triangle, label the hypotenuse first, and check that the answer is sensible: the hypotenuse must be longer than either other side. Problems are often set in context, such as a ladder against a wall, the diagonal of a rectangle, or the distance between two points on a coordinate grid.

Trigonometry starts from similar triangles. If two right-angled triangles have one other angle the same, all their angles match and they are similar, so the ratios of their sides match. For an angle of 30°, the side opposite it is always half the hypotenuse, whatever the triangle’s size. The programme of study asks pupils to use trigonometric ratios in similar triangles to solve problems with right-angled triangles. Some schools introduce sine, cosine and tangent formally in Year 9, with the ratios written opposite ÷ hypotenuse (sine), adjacent ÷ hypotenuse (cosine) and opposite ÷ adjacent (tangent). Others spend more time on similar triangles and leave the notation for GCSE, so the pace differs between schools. The hypotenuse is always the side opposite the right angle, but which side is “opposite” and which is “adjacent” depends on the angle being used.

Geometric reasoning ties these ideas together. Pupils use angle facts, congruence, similarity and the properties of quadrilaterals to derive new results and write simple proofs, with each step supported by a reason. They might prove that the angles of a quadrilateral add up to 360°, that the base angles of an isosceles triangle are equal, or that the diagonals of a rhombus meet at right angles. A classic proof of Pythagoras’ Theorem rearranges four copies of a right-angled triangle inside a square.

Finally, pupils are asked to interpret relationships both algebraically and geometrically. For example, a² + b² = c² is an equation, but it also describes areas of squares. The equation y = 2x is a rule linking two numbers, but it is also a straight line on a graph that rises 2 units for each 1 across. Moving between the picture and the algebra is the habit being built here.

Words your child will hear

Hypotenuse: The longest side of a right-angled triangle, always opposite the right angle.
Pythagoras’ Theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse.
Square root: The number that, multiplied by itself, gives a number. The square root of 25 is 5.
Opposite and adjacent: The side facing an angle, and the side next to it (not the hypotenuse).
Sine, cosine, tangent: Ratios of two sides of a right-angled triangle that depend only on the angle.
Similar triangles: Triangles with equal angles whose sides are in the same ratio.
Proof: A chain of logical steps, each with a reason, that shows a result is always true.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs
Use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles
Interpret mathematical relationships both algebraically and geometrically

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

A worked example

A ladder 5 m long leans against a vertical wall. Its foot is 1.5 m from the wall on level ground. How high up the wall does the ladder reach? Give your answer to 3 significant figures.

Sketch it. The wall is vertical and the ground is horizontal, so there is a right angle. The ladder is the hypotenuse, 5 m. One shorter side is 1.5 m, and the height h is the other.
Write Pythagoras’ Theorem with the hypotenuse on its own side.
```
1.5² + h² = 5²
```

We are finding a shorter side, so subtract.

2.25 + h² = 25
      h² = 25 − 2.25
      h² = 22.75

Take the square root to find h. Do not stop at 22.75, which is the square of the height.
```
h = √22.75
  = 4.77 m (3 s.f.)
```
Check: the height is less than the ladder’s length, as it must be, and 4.77² is about 22.75.

The ladder reaches about 4.77 m up the wall.

Common mistakes and how to help

Adding when a shorter side is needed

Pythagoras is often learned as “square, add, square root”, so pupils do it every time, even when the hypotenuse is the known side.

What helps: Ask first: is the missing side the longest? If so, add. If it is one of the shorter sides, subtract. A sketch with the hypotenuse labelled c settles it.

Stopping at c² or squaring after adding

Calculating 3² + 4² = 25 feels like an answer, and some pupils write (3 + 4)² instead, which is 49.

What helps: Say the steps out loud: square each, add, then square root. The answer must be a length, so 25 is not the side. Use the 3, 4, 5 triangle as a check.

Labelling opposite and adjacent from the wrong angle

Opposite and adjacent are not fixed names for sides. A side that is opposite one angle is adjacent to the other.

What helps: Mark the angle in the question first, then label the hypotenuse, then the opposite side facing the angle. Whatever is left is adjacent. Make sure the calculator is in degree mode.

Using Pythagoras on a triangle without a right angle

The formula looks so familiar that pupils apply it to any triangle with three sides.

What helps: Look for the right-angle mark or a clear reason the angle must be 90°. If there is none, Pythagoras alone will not work.

How to help at home

Build a 3, 4, 5 corner. Mark 3 units along one wall and 4 along another from a corner. If the distance between the marks is exactly 5, the corner is a right angle. Use a ruler, a tape or lengths of string.
Find diagonals. Measure a rectangular tray, a phone screen or a table top and use Pythagoras to predict the diagonal. Then measure it to see how close you got.
Compare shadows. On a sunny day, measure your shadow and a tall object’s shadow. The triangles formed are similar, so the ratios match, and you can estimate the object’s height. Do it in a garden or park, and remind your child never to look directly at the sun.
Talk through a proof. Pick a fact, such as angles in a triangle summing to 180°, and ask your child to explain why it must be true, step by step, with a reason for each.

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