Cuboid Perimeter and Volume

Let's say the cuboid has sides of lengths $a, b$ and $c$ . The perimeter of each of the 3 main faces are $2a + 2b, 2b +2c, 2c + 2a$ , arbitrarily assign them to the faces given (since it is a cuboid any permutation of the faces will end with the same result)

Face 1: $2a + 2b = 6 \Rightarrow a + b = 3 \Rightarrow a = 3 - b$
Face 2: $2b + 2c = 8 \Rightarrow b + c = 4 \Rightarrow c = 4 - b$
Face 3: $2c + 2a = 10 \Rightarrow c + a = 5$

Substituting the first two equations into the third, we get $\begin{aligned} 3-b + 4-b &= 5 \\ 7 - 2b &= 5 \\ 2b &= 2 \\ b&=1 \end{aligned}$

Hence, $c = 4-b = 4-1 = 3$ and $a = 3-b = 3-1 = 2$ .
Thus, the Volume $= abc = 1 * 2 * 3 = 6.$