What Did I Miss?

By Heron's formula ,

$\text{Area} = \frac {1}{4} \sqrt{ \left (P^2 + Q^2 + R^2 \right )^2 - 2\left (P^4 + Q^4 + R^4 \right ) } = \boxed{12}$

$p = \frac{P+Q+R}{2}$ $S = \sqrt{p(p-P)(p-Q)(p-R)}$ $S^2 = \left (\frac{P+Q+R}{2} \right )\left (\frac{P+Q-R}{2} \right )\left (\frac{P-Q+R}{2} \right )\left (\frac{-P+Q+R}{2} \right )$ $16S^2 = (P+Q+R)(P+Q-R)(P-Q+R)(-P+Q+R)$ $16S^2 = 2(P^2Q^2 + P^2R^2 + Q^2R^2) - (P^4+Q^4+R^4)$ $16S^2 = (P^2 + Q^2 + R^2)^2 - 2(P^4 + Q^4 + R^4)$ $16S^2 = 114^2 - 2 \cdot 5346$ $16S^2 = 2304$ $\boxed{S = 12.}$