Calculating Surface Area of Composite 3D Figure
To find the surface area of the 3D figure provided in the image, you need to calculate the area of each visible surface and then add them all together.
The figure is a composite solid made up of two rectangular prisms. We must be careful not to double-count any area that is shared between the two prisms.
1. The large, vertical front face is 16 in. by 16 in., so its area is \(16\ in \times 16\ in = 256\ in^2\).
2. The top face of the larger prism is 16 in. by 8 in., so its area is \(16\ in \times 8\ in = 128\ in^2\).
3. The back face of the larger prism, which is identical to the front, is also \(16\ in \times 16\ in = 256\ in^2\).
4. The bottom face of the larger prism is identical to the top face and has an area of \(16\ in \times 8\ in = 128\ in^2\).
5. The left face of the large prism is 16 in. by 8 in., giving an area of \(16\ in \times 8\ in = 128\ in^2\).
Now let's consider the smaller, attached prism:
6. The visible part of the right side of the larger prism is an 8 in. by 8 in. square (above the smaller prism), with an area of \(8\ in \times 8\ in = 64\ in^2\).
7. The front face of the smaller prism is 6 in. by 8 in., giving an area of \(6\ in \times 8\ in = 48\ in^2\).
8. The top face of the smaller prism (visible portion) is 6 in. by 8 in., with an area of \(6\ in \times 8\ in = 48\ in^2\).
9. The right-side face of the smaller prism (aligned with the front face of the larger prism) is 6 in. by 16 in., with an area of \(6\ in \times 16\ in = 96\ in^2\).
Now let's add up all these areas to find the total surface area:
\(256\ in^2 + 128\ in^2 + 256\ in^2 + 128\ in^2 + 128\ in^2 + 64\ in^2 + 48\ in^2 + 48\ in^2 + 96\ in^2 = 1152\ in^2\).
So the total surface area of the figure is 1,152 square inches.
