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Pythagoras' theorem

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Pythagoras' theorem

In a nutshell

Pythagoras' theorem is very useful when trying to find the value for the third side of a right-angled triangle. Additionally, at this level, you may have to apply the formula to finding the distance between 2 points on a graph.


Pythagoras' theorem says that:


Where aa​ and bb​ are the two shorter sides of a right-angled triangle and cc​ is the hypotenuse.

Note: Pythagoras' theorem only works for a right-angled triangle. If there is no right-angle, then you cannot use Pythagoras' theorem.


The hypotenuse of a triangle is the longest side. For a right-angled triangle, the hypotenuse is always the side opposite the right-angle.

Distance between two points on a graph

You may be asked to find the distance between 2 given coordinates. Approaching these types of problems always requires the same method.



Do a quick sketch of the graph with the given coordinates and draw a right-angled triangle with the two points.


Find the lengths of the two shorter sides of the triangle.


Use Pythagoras' theorem to find the length of the hypotenuse of the triangle - this is the distance.


Find the distance between the points A(1,3)A (-1,3)​ and B(5,5)B(5,-5).

First, make a quick sketch:

Maths; Trigonometry; KS4 Year 10; Pythagoras' theorem

The length of the horizontal side is:

5(1)=65 - (-1) = 6

The length of the vertical side is:

3(5)=83 - (-5) = 8

The distance (labelled dd) will then satisfy Pythagoras' theorem:

62+82=d26^2 + 8^2 = d^2​​


d2=100d^2 = 100​​

d=100=10\underline{d = \sqrt{100} = 10}

Want to find out more? Check out these other lessons!

Pythagoras’ theorem

Frequently Asked Questions (FAQ)


  • Question: How do I find the distance between 2 points on a graph?

    Answer: First sketch the graph and points. Then, make a right-angled triangle with the points and use Pythagoras' Theorem.

  • Question: What is the hypotenuse of a triangle?

    Answer: The hypotenuse of a triangle is the longest side.

  • Question: When can I use Pythagoras' Theorem?

    Answer: You can only use Pythagoras' Theorem when you are given 2 sides of a right-angled triangle and want to find the third side.



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