A hemispherical bowl of radius 4 cm is cut into the top face of a cube of edge 10 cm…

Step 1. The plain cube. $6a^2 = 6 \times (10\ \text{cm})^2 = 600\ \text{cm}^2$.

Step 2. The circle removed. $\pi r^2 = 16\pi\ \text{cm}^2$.

Step 3. The bowl added. $2\pi r^2 = 32\pi\ \text{cm}^2$.

Step 4. The total. $600 - 16\pi + 32\pi = 600 + 16\pi\ \text{cm}^2 \approx 650.3\ \text{cm}^2$.