# Smallest Scalene Obtuse Triangle with Integer Sides and Area

## Theorem

The smallest scalene obtuse triangle with integer sides and area has sides of length $4, 13, 15$.

## Proof

From Heron's Formula, the area $A$ of $\triangle ABC$ is given by:

$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$

where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.

Here we have:

 $\ds s$ $=$ $\ds \dfrac {4 + 13 + 15} 2$ $\ds$ $=$ $\ds 16$

Thus:

 $\ds A$ $=$ $\ds \sqrt {16 \paren {16 - 4} \paren {16 - 13} \paren {16 - 15} }$ $\ds$ $=$ $\ds \sqrt {16 \times 12 \times 3 \times 1}$ $\ds$ $=$ $\ds \sqrt {2^4 \times 2^2 \times 3 \times 3 \times 1}$ $\ds$ $=$ $\ds \sqrt {2^6 \times 3^2}$ $\ds$ $=$ $\ds 2^3 \times 3$ $\ds$ $=$ $\ds 24$