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Pythagoras' Theorem

FoundationHigherAQAEdexcelOCR

Practise Pythagoras' theorem with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through using Pythagoras’ theorem to find missing sides, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Pythagoras only works in right-angled triangles: a² + b² = c².

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

Pythagoras' theorem connects the three sides of a right-angled triangle. It states that the square of the hypotenuse equals the sum of the squares of the other two sides, written \(a^2 + b^2 = c^2\).

The hypotenuse is always the longest side and always sits opposite the right angle. Identifying it correctly is the first thing to do in any question, because the formula treats it differently from the other two sides.

The theorem works in both directions. If you know two sides you can find the third, and if you know all three you can test whether the triangle contains a right angle. It also underpins later work on coordinates, vectors and three-dimensional shapes.

Revision notes

Finding the hypotenuse

When the missing side is the longest one, square both known sides, add them, and take the square root.

For legs of \(3\) and \(4\): \(3^2 + 4^2 = 9 + 16 = 25\), so \(c = \sqrt{25} = 5\). Adding is the giveaway that you are finding the hypotenuse.

Finding a shorter side

When the missing side is one of the shorter ones, you subtract instead. Square the hypotenuse, subtract the square of the known side, then take the root.

With a hypotenuse of \(13\) and one leg of \(5\): \(13^2 - 5^2 = 169 - 25 = 144\), so the other leg is \(12\). Adding here would give an impossible answer larger than the hypotenuse.

Testing for a right angle

Substitute all three sides into \(a^2 + b^2 = c^2\), putting the longest side as \(c\). If both sides of the equation are equal, the triangle is right-angled.

For \(6\), \(8\) and \(10\): \(36 + 64 = 100\), which matches \(10^2\), so it is right-angled.

Key points

  • Applies only to right-angled triangles.
  • \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
  • The hypotenuse is opposite the right angle and is the longest side.
  • Add the squares to find the hypotenuse.
  • Subtract the squares to find a shorter side.
  • Take the square root at the end, not before.

Worked examples

Example 1

A right-angled triangle has legs of \(9\ \text{cm}\) and \(12\ \text{cm}\). Find the hypotenuse.

Working

\[9^2 + 12^2 = 81 + 144\]square both shorter sides and add, since the hypotenuse is missing
\[= 225\]add the squares
\[\sqrt{225} = 15\ \text{cm}\]take the square root

Example 2

A ladder \(5\ \text{m}\) long leans against a wall with its foot \(1.4\ \text{m}\) from the base. Find how far up the wall it reaches.

Working

\[5^2 - 1.4^2 = 25 - 1.96\]the ladder is the hypotenuse, so subtract to find a shorter side
\[= 23.04\]work out the difference
\[\sqrt{23.04} = 4.8\ \text{m}\]take the square root

Example 3

Show that a triangle with sides \(7\), \(24\) and \(25\) is right-angled.

Working

\[7^2 + 24^2 = 49 + 576\]square the two shorter sides and add
\[= 625\]work out the total
\[25^2 = 625 \text{, so it is right-angled}\]compare with the square of the longest side

Common mistakes

  • Adding when the missing side is a shorter one.

    This gives an answer longer than the hypotenuse, which is impossible. Subtract when the hypotenuse is known.

  • Forgetting to take the square root.

    Stopping at 225 gives the square of the answer, not the length. The final step is always the root.

  • Using it on a triangle with no right angle.

    Pythagoras applies only to right-angled triangles. Use the sine or cosine rule otherwise.

  • Mixing units within one calculation.

    Convert everything to the same unit before squaring, or the totals are meaningless.

Exam tips

  • Label the hypotenuse first, before writing anything down.
  • Sense-check your answer: the hypotenuse must be the longest side.
  • Keep full accuracy in the working and round only at the end.
  • Include units in the final answer, since geometry questions expect them.

Key terms

Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Pythagoras' theorem
The rule that \(a^2 + b^2 = c^2\) in a right-angled triangle.
Square root
The value that multiplies by itself to give a number.
Right angle
An angle of 90 degrees.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.