Pythagoras' Theorem

What Pythagoras' theorem connects

Pythagoras' theorem links the three side lengths of a right-angled triangle. If the two shorter sides have lengths $a$ and $b$, and the hypotenuse has length $c$, then

$a^2+b^2=c^2.$

The letters do not name fixed positions on the page. Either shorter side may be $a$ or $b$, but $c$ must be the hypotenuse. Find the right-angle marker first: the side directly opposite it is the hypotenuse. It is also the triangle's longest side.

The equation compares squared lengths. It says that the area of a square drawn on one shorter side plus the area of a square drawn on the other shorter side equals the area of a square drawn on the hypotenuse. This is why the equation contains squares and why a square root is needed to recover an ordinary length.

Although the theorem is named after the ancient Greek mathematician Pythagoras, the important condition is mathematical: it applies only when the triangle is right-angled.

Finding the hypotenuse

Suppose the two shorter sides are known. Rearrange the theorem for $c$:

$c=\sqrt{a^2+b^2}.$

The process is therefore: square both shorter lengths, add the squares, and take the positive square root. A length is positive, so the positive square root is used.

For shorter sides of $6\text{ cm}$ and $8\text{ cm}$,

$c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10\text{ cm}.$

The answer is sensible because $10\text{ cm}$ is longer than both known sides.

Finding a shorter side

If the hypotenuse and one shorter side are known, begin with the same theorem and rearrange it. For example, to find $a$:

$a^2=c^2-b^2$

and hence

$a=\sqrt{c^2-b^2}.$

For a hypotenuse of $13\text{ m}$ and a known shorter side of $5\text{ m}$,

$a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12\text{ m}.$

Subtraction is used because the hypotenuse's square already equals the sum of the other two squares. Always subtract the square of the shorter side from the square of the hypotenuse, so the value inside the square root is non-negative.

Recognising a hidden right-angled triangle

A problem may not present an isolated triangle. A rectangle's diagonal, for example, divides it into two right-angled triangles. The diagonal is opposite the right angle and is therefore the hypotenuse. Perpendicular horizontal and vertical distances can form the same structure in a coordinate diagram.

In a word problem, sketch the situation and label the right angle. A ladder leaning against a vertical wall forms a right-angled triangle with the wall and level ground: the ladder is the hypotenuse. Turning the situation into a labelled triangle prevents the measurements from being assigned to the wrong sides.

Some diagrams require more than one step. One right-angled triangle may first provide an intermediate length, which is then used in another. Keep an exact root in the working rather than replacing it with an early rounded decimal. This prevents rounding error from accumulating.

Presenting and checking the result

Show the general formula, substitute the side lengths, rearrange if necessary, and then take the square root. Give the final length with its unit and round only as requested.

A quick check can expose mistakes. The hypotenuse must be the longest side, so a calculated hypotenuse shorter than a known side cannot be correct. When finding a shorter side, its length must be less than the known hypotenuse. Remember also that $a^2+b^2$ or $c^2-b^2$ is a squared length; the final square root is essential.

Identify the sides

Pythagoras' theorem applies to a . The is opposite the right angle, is the longest side and must be labelled .