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What is pythagoras' theorem?

Pythagoras' theorem states that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides.

Pythagoras' theorem explained

Use it to find a missing side when you know the other two and the triangle contains a right angle. Rearrange by subtracting when you are finding a shorter side.
It extends to three dimensions for the diagonal of a cuboid, and to coordinates for the distance between two points.

Key formula

a2+b2=c2a^{2} + b^{2} = c^{2}

Worked example

With legs 66 and 88: c=36+64=10c = \sqrt{36 + 64} = 10.

Examiner tip

Check the right angle exists before using it — if there is no right angle, the question wants the cosine rule instead.

Pythagoras' theorem: common questions

Can Pythagoras be used on any triangle?
No — only right-angled triangles. For other triangles use the sine or cosine rule.
How do I know if a triangle is right-angled?
Test whether the squares of the two shorter sides add to the square of the longest side; if they do, the triangle is right-angled.

Related terms

Trigonometry

Trigonometry relates the angles of a triangle to the lengths of its sides using the sine, cosine and tangent ratios.

Vector

A vector is a quantity with both magnitude and direction, such as displacement, velocity or force.

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