Cubes on a Cube

The shape consists of $7$ cubes, and has a total volume of $875\text{cm}^3$. Each cube therefore has volume $875\text{cm}^3 \div 7 = 125\text{cm}^3$.

Therefore, the side length of the cube is $\sqrt[3]{125\text{cm}^3} = 5\text{cm}$, so each face has area $(5\text{cm})^2 = 25\text{cm}^2$.

Each of the outer cubes has five of its faces showing, so there are $6 \times 5 = 30$ faces showing altogether. These have a total area of $30 \times 25\text{cm}^2 = 750\text{cm}^2$.