Building Jack's Prism

Let $a$ and $b$ be the lengths of the rectangular base and $h$ be the height of the prism. $a$ , $b$ and $h$ must be positive integers. Then $2a + 2b = 10 \Rightarrow a + b = 5$ and $abh = 222 = 2\times 3 \times 37$ . Furthermore, $a$ and $b$ are factors of $222$ that sum to $5$ , of which only $2$ and $3$ work. Thus, these are the lengths of the rectangular base, which allows us to calculate the height as $h = \frac{2\times 3 \times 37}{2 \times 3} = 37$ .