The formula a² + b² = c² is memorable enough to be used too quickly. Before substituting any numbers, look for the condition that makes it applicable: a right-angled triangle. The letters are labels; the right angle is the mathematical requirement.
OpenStax’s introduction to the Pythagorean theorem explains the relationship between the two legs and the hypotenuse of a right triangle. The hypotenuse is opposite the right angle and is the longest side. It remains the hypotenuse when the diagram is rotated.
Mark the sides before calculating
Consider an invented example with perpendicular sides measuring 5 cm and 12 cm. The unknown diagonal is opposite the right angle, so it is c:
c² = 5² + 12² = 25 + 144 = 169c = √169 = 13 cm
The squared lengths produce square centimetres during the calculation. Taking the positive square root gives a length in centimetres. A negative length is not the other answer to this geometry problem, even though an algebraic equation may have two signed roots.
What if the missing side is a leg?
Suppose the hypotenuse is 13 cm and one leg is 5 cm. The remaining leg satisfies b² = 13² − 5² = 144, so it measures 12 cm. Adding the two given squares here would assign the wrong role to the unknown side.
Before accepting an answer, ask whether it makes sense. The hypotenuse must be longer than either leg. All lengths must use consistent units. The final line should give the side length, not stop at its square.
Do not assume a right angle merely because a sketch looks nearly square. Use the information supplied in the problem, such as an angle label or a stated perpendicular relationship. For a triangle without that condition, another method may be needed.
A useful habit is to circle the right angle, label the opposite side and only then write the equation. That short pause prevents more errors than memorising the formula alone.