Question - Calculating the Surface Area of a Combined Solid

Let us denote the following:

\( A_p \) = The area of the pyramid base

\( A_c \) = The area of the cube

\( A_{t} \) = The area of one triangle on the pyramid

\( A_{\text{total}} \) = The total surface area

First, we calculate the base area of the pyramid (square):

\[ A_p = 5 \times 5 = 25 \ \text{cm}^2 \]

Then, the area of one of the four identical triangles of the pyramid:

\[ A_{t} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 6 = 15 \ \text{cm}^2 \]

Now, the surface area of the cube minus the area where the pyramid is attached:

\[ A_c = 6 \times \text{face area} - A_p = 6 \times (5 \times 5) - 25 = 150 \ \text{cm}^2 \]

Adding the four triangles' area:

\[ A_c = A_c + 4 \times A_{t} = 150 + 4 \times 15 = 210 \ \text{cm}^2 \]

The combined surface area:

\[ A_{\text{total}} = A_c + A_p = 210 + 25 = 235 \ \text{cm}^2 \]