The Pythagorean Theorem with Worked Examples

In any right triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides: $a^2 + b^2 = c^2$. Use it to find a missing side, to test whether a triangle has a right angle, and to find the distance between two points.

The theorem

Here $c$ is always the hypotenuse, the side opposite the right angle. Getting this wrong is the most common mistake.

Worked example 1: find the hypotenuse

The shorter sides are 3 and 4.

Worked example 2: find a shorter side

The hypotenuse is 13 and one side is 5. Rearrange:

Worked example 3: an irrational answer

Sides 5 and 7:

$74 = 2 \times 37$ has no square factors, so $\sqrt{74}$ is already in simplest form.

Pythagorean triples

Whole-number solutions are called Pythagorean triples. Learning a few saves time:

Is it a right triangle? (the converse)

If $a^2 + b^2 = c^2$ for the longest side $c$, the triangle has a right angle. If $a^2 + b^2 > c^2$ it is acute; if $a^2 + b^2 < c^2$ it is obtuse.

Example: sides 6, 8, 11. $36 + 64 = 100 < 121$, so the triangle is obtuse.

Distance between two points

The horizontal and vertical gaps between two points are the shorter sides of a right triangle. From $(1, 2)$ to $(4, 6)$ the gaps are $3$ and $4$:

That is the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Common mistakes

• Using a shorter side as $c$.
• Adding the sides before squaring: $(3 + 4)^2 \ne 3^2 + 4^2$.
• Forgetting to take the square root at the end.

Practice questions

1. Find the hypotenuse when the other sides are 8 and 15.
2. Find the missing side when the hypotenuse is 25 and one side is 7.
3. Is a triangle with sides 9, 12, 15 right-angled?

Answers: 1) 17 2) 24 3) Yes, $81 + 144 = 225 = 15^2$

Find any side with the Pythagorean theorem calculator, or the gap between two points with the distance calculator. For simplifying roots like $\sqrt{72}$, see how to simplify square roots.

Frequently asked questions

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