GCSE Maths · Geometry & measures

Pythagoras

Pythagoras' theorem in right-angled triangles - finding the hypotenuse or a shorter side, and (Higher) in 3D.

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Pythagoras, explained point by point

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  1. State Pythagoras' theorem and know when it applies

    In a right-angled triangle, a² + b² = c², where c is the hypotenuse - the longest side, opposite the right angle. It links the three SIDES and needs no angles; if there is no right angle, it does not apply (that is cosine-rule territory).

  2. Find the hypotenuse of a right-angled triangle

    Square the two shorter sides, add, square-root: legs of 6 and 8 give 36+64 = 100 = 10. The hypotenuse must come out LONGER than either leg - a quick sanity check.

  3. Find a shorter side of a right-angled triangle

    Rearrange to subtract: b² = c² − a². With hypotenuse 13 and one leg 5: 169−25 = 144 = 12. If you catch yourself adding when the hypotenuse is known, the answer will come out impossibly long.

  4. Decide whether a triangle is right-angled from its sides

    Test whether the two shorter sides' squares sum to the longest side's square. For 20, 21, 29: 400 + 441 = 841 = 29², so yes. Show the arithmetic and state the conclusion - that is the converse of Pythagoras in action.

  5. Find the distance between two points on a coordinate grid

    The horizontal and vertical gaps between the points form the legs of a right-angled triangle, and the distance is its hypotenuse. From (1, 2) to (7, 10): gaps of 6 and 8, so the distance is 36+64 = 10.

  6. Use Pythagoras' theorem in 3D problemsHigher only

    Apply it twice: once across the base to get a diagonal, then again using that diagonal and the height. A 3-4-12 box has base diagonal 9+16 = 5, then space diagonal 25+144 = 13. Sketch each right-angled triangle flat before calculating.

Practice

Try a Pythagoras question

A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.

Work out the distance between the points A(1, 2) and B(4, 6). Give your answer to 1 decimal place.

  1. 1.0
  2. 5.0
  3. 25.0
  4. 7.0
Show the answer and the working

Answer: 5.0

The distance between two points is x₂−x₁2+y₂−y₁2, the hypotenuse of the horizontal and vertical gaps.

  1. Horizontal gap: 4 − 1 = 3. Vertical gap: 6 − 2 = 4.
  2. These are the legs of a right-angled triangle; the distance is the hypotenuse.
  3. Distance = 32+42 = 25 = 5.0.

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