Calculating the hypotenuse

Pythagoras’ theorem allows us to calculate the length of any side of a right-angled triangle given the other two.

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The square of the hypotenuse is equal to the sum of the squares of the remaining two sides.Pythagoras’ theorem

The hypotenuse is the longest side – it will always be opposite the right angle.

Three right-angled triangles with an arrow pointing to the hypotenuse

To represent this in a mathematical formula we can say;

{a}{^2}~=~{b}{^2}~{+}~{c}{^2}

Where a is the length of the hypotenuse and the other sides are labelled b and c.

Right-angled triangle with sides a, b and c, where a is the hypotenuse

In this triangle we need to find the hypotenuse.

Right-angled triangle with sides of length 3cm, 4cm, and x, which is the hypotenuse

Pythagoras’ theorem tells us that:

{x}{^2}~=~{3}{^2}~{+}~{4}{^2}

{x}{^2}~=~{9}~{+}~{16}

{x}{^2}~=~{25}

To find {x}, we need to square root both sides added together.

{x} = \sqrt{25}

{x}~{=}~{5}~{cm}

Question

Find the length AC, giving your answer to two decimal places.

Right-angled triangle where the side AB equals 6m, BC equals 2m, and AC is the hypotenuse

AC2 = 62 + 22

AC2 = 36 + 4

AC2 = 40

AC = \sqrt{40}

AC2 = 6.32455532

AC2 = 6.32 m (to two decimal places)