Pythagorean Theorem — Formula and Exercises

The Pythagorean theorem relates the three sides of a right triangle. With legs $a$ and $b$ and hypotenuse $c$:

The symbols carry the conditions. The legs $a$ and $b$ are the two sides that form the right angle; the hypotenuse $c$ is the side opposite that angle and is therefore always the longest side. The relationship applies only when the triangle has an angle of $90^\circ$ — without it, using the theorem is simply incorrect.

Why the side identification controls the result

The letter $c$ in the formula represents the hypotenuse specifically. Because the theorem balances the square on the longest side against the squares on the two shorter ones, swapping any side into the role of $c$ sets up the equation incorrectly from the start. So before any arithmetic, fix which side is the hypotenuse and which two are the legs.

This also gives you two rearrangements that all follow from the one identity. To find the hypotenuse from both legs:

To isolate a leg when you know the hypotenuse and one leg:

These only make sense when $c$ is genuinely the hypotenuse and larger than the other known side.

Worked example: finding the hypotenuse

A right triangle has legs of $6$ cm and $8$ cm. Find the hypotenuse. Apply the formula:

The hypotenuse measures $10$ cm. It passes a quick check: the result is larger than both $6$ and $8$, as it must be in any right triangle.

Try the calculation yourself

Practice 1 — finding the hypotenuse. Use legs of $9$ cm and $12$ cm. Set up $a^2 + b^2 = c^2$, then confirm the answer exceeds both legs.

Practice 2 — finding a leg. Suppose the hypotenuse is $13$ cm and one leg is $5$ cm:

The trap here is forgetting the square root: since $a^2 = 144$, the side is $a = 12$, not $144$. For one more step, try a leg when the hypotenuse is $15$ cm and the other side is $9$ cm.

Calculation traps to avoid

• Using the formula on any triangle. The essential condition is a right angle; without it the relationship does not hold.
• Confusing a leg with the hypotenuse. The hypotenuse is always the longest side; mistaking a leg for it sets up the equation wrong.
• Forgetting the square root at the end. When the work reaches $c^2 = 169$, the side is $c = 13$, not $169$.

The theorem shows up in plane geometry, distance problems, civil construction, map navigation, and diagonal analysis — for instance, the diagonal of a rectangle with sides $9$ and $12$ is the hypotenuse of a right triangle. In each case it turns a drawing into an objective calculation.