The right triangle whose perimeter equals its area

Text

Consider the right triangle with legs $5$ and $12$. Check that it is Pythagorean, then compute its perimeter and its area and note the coincidence.

Solution

The hypotenuse is $\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$: the triple $5,12,13$ is Pythagorean.

• Perimeter: $5+12+13 = 30$.
• Area: $\dfrac{5\cdot 12}{2} = 30$.

Perimeter and area are both $\boxed{30}$: a rare “perfect” triangle where the two numbers coincide.