The Pythagorean Theorem

The Pythagorean theorem finds a missing side of a right-angled triangle. If $a$ and $b$ are the sides forming the right angle and $c$ is the hypotenuse, then $a^2 + b^2 = c^2$.

The formula and its conditions

The theorem says the square of the longest side equals the sum of the squares of the other two. Two conditions control whether it applies: the triangle must be a right triangle, and $c$ must be the side opposite the right angle. So before any calculation, ask a single question: is there a right angle? If not, this relationship does not hold in this form.

Why the hypotenuse label matters

In a right triangle the hypotenuse is always the longest side, so it is the side that must be labeled $c$. Many mistakes come from a correct calculation done with the wrong variables. If you label a shorter side as $c$, the algebra can look fine even though the starting model is wrong.

Worked example: find the hypotenuse

Take a right triangle with $a = 6$ and $b = 8$, and look for the hypotenuse $c$.

Start with

Substitute the values:

Take the positive square root, because a length must be positive:

The result is consistent: $10$ is larger than both $6$ and $8$, so it can indeed be the hypotenuse.

Practice it yourself, then check

Now try sides $5$ and $12$.

Check: $5^2 + 12^2 = 25 + 144 = 169$, and $\sqrt{169} = 13$. If you found $13$, you have not just memorized the formula, you have understood how to use it correctly, including taking the square root at the end and confirming $13$ is the longest side.

Calculation traps

• Using the formula in a triangle that is not a right triangle. Without a right angle the relationship no longer holds.
• Forgetting that $c$ is the hypotenuse. The correct formula with the wrong sides gives an incorrect result.
• Stopping at $c^2 = 100$ without taking the square root. The length $c$ is $10$, not $100$.
• Failing to check whether the answer makes sense. The hypotenuse must always remain the longest side.

When to use the Pythagorean theorem

Use it whenever a problem involves a right angle and a missing length: diagonals of a rectangle, certain distance problems, and work in an orthogonal coordinate system. For example, if a movement consists of $3$ units horizontally and then $4$ units vertically, the direct distance is

This same idea underlies the distance formula in a plane when the axes are perpendicular. Two quick questions settle most cases before you calculate: is there a right angle, and have I correctly identified the hypotenuse?