Given:
• Length of base(cube) = 6 ft
,
• Height of square prism = 20 ft
,
• Height of pyramid =4 ft
Since it is a composite figure, the surface of some sides will not be included in the surface area.
Let's solve for the following:
• (a). The surface area of the pyramid.
To find the surface area of the pyramid, apply the formula:
![SA=A+\frac{1}{2}ps](https://tex.z-dn.net/?f=SA%3DA%2B%5Cfrac%7B1%7D%7B2%7Dps)
Where:
A is the area of the base
p is the perimeter of the base
s is the slant height.
To find the slant height, apply Pythagorean theorem:
![\begin{gathered} s=\sqrt{(4)^2+(\frac{6}{2})^2} \\ \\ s=\sqrt{16+9}=\sqrt{25}=5\text{ ft} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20s%3D%5Csqrt%7B%284%29%5E2%2B%28%5Cfrac%7B6%7D%7B2%7D%29%5E2%7D%20%5C%5C%20%20%5C%5C%20s%3D%5Csqrt%7B16%2B9%7D%3D%5Csqrt%7B25%7D%3D5%5Ctext%7B%20ft%7D%20%5Cend%7Bgathered%7D)
We have:
![\begin{gathered} SA=(6*6)+\frac{1}{2}(6*4)*5 \\ \\ SA=36+\frac{1}{2}(24)*5 \\ \\ SA=36+60 \\ \\ SA=96\text{ ft}^2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20SA%3D%286%2A6%29%2B%5Cfrac%7B1%7D%7B2%7D%286%2A4%29%2A5%20%5C%5C%20%20%5C%5C%20SA%3D36%2B%5Cfrac%7B1%7D%7B2%7D%2824%29%2A5%20%5C%5C%20%20%5C%5C%20SA%3D36%2B60%20%5C%5C%20%20%5C%5C%20SA%3D96%5Ctext%7B%20ft%7D%5E2%20%5Cend%7Bgathered%7D)
The surface area of the pyramid is 96 ft².
• (b). Surface area of the square prism.
Apply the formula:
![\begin{gathered} SA=3ah \\ \\ SA=4(6)(20) \\ \\ SA=480\text{ ft}^2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20SA%3D3ah%20%5C%5C%20%20%5C%5C%20SA%3D4%286%29%2820%29%20%5C%5C%20%20%5C%5C%20SA%3D480%5Ctext%7B%20ft%7D%5E2%20%5Cend%7Bgathered%7D)
The surface area of the square prism is 480 ft².
• (c). Surface area of the cube.
To find the surface area of the cube, we have:
![\begin{gathered} SA=6a^2 \\ \\ SA=6(6)^2 \\ \\ SA=216\text{ ft}^2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20SA%3D6a%5E2%20%5C%5C%20%20%5C%5C%20SA%3D6%286%29%5E2%20%5C%5C%20%20%5C%5C%20SA%3D216%5Ctext%7B%20ft%7D%5E2%20%5Cend%7Bgathered%7D)
The surface area of the cube is 216 ft²
The total surface area will be:
T = Surface area of pyramid + surface area of prism + surface area of cube - area of base
T = 96 + 480 + 216 - 36
T = 756 ft²
ANSWER:
Surface area of the pyramid is 96 ft².
Surface area of the square prism is 480 ft².
Surface area of the cube is 216 ft²
Total surface area is 756 ft²