Example Question - combined solids

Calculating the Surface Area of a Combined Solid

<p>Let us denote the following:</p> <p>\( A_p \) = The area of the pyramid base</p> <p>\( A_c \) = The area of the cube</p> <p>\( A_{t} \) = The area of one triangle on the pyramid</p> <p>\( A_{\text{total}} \) = The total surface area</p> <p>First, we calculate the base area of the pyramid (square):</p> <p>\[ A_p = 5 \times 5 = 25 \ \text{cm}^2 \]</p> <p>Then, the area of one of the four identical triangles of the pyramid:</p> <p>\[ A_{t} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 6 = 15 \ \text{cm}^2 \]</p> <p>Now, the surface area of the cube minus the area where the pyramid is attached:</p> <p>\[ A_c = 6 \times \text{face area} - A_p = 6 \times (5 \times 5) - 25 = 150 \ \text{cm}^2 \]</p> <p>Adding the four triangles' area:</p> <p>\[ A_c = A_c + 4 \times A_{t} = 150 + 4 \times 15 = 210 \ \text{cm}^2 \]</p> <p>The combined surface area:</p> <p>\[ A_{\text{total}} = A_c + A_p = 210 + 25 = 235 \ \text{cm}^2 \]</p>