An illustration shows a three dimensional composite figure formed by a right triangular prism placed on top of a rectangular prism. The top, right, and front faces are visible. The faces and edges that are not visible are indicated by dashed lines. The rectangular prism has a length of 12 feet and a height of 8 feet. The width of the rectangular prism is 8 feet. The edges of the hidden side face of the right triangular prism align perfectly with the edges of the hidden top face of the rectangular prism. The right triangular prism is aligned to the left of the rectangular prism. The height of the right triangular prism is not shown. The total height of the left side of the figure is 20 feet. The right triangular prism has a width of 8 feet. The hypotenuse side of the right triangular prism has a length of 15 feet.
What is the surface area of the figure?
To find the surface area of the figure, we need to find the surface area of each individual shape and then add them together.
Surface area of rectangular prism:
- Front and Back faces: 12 ft (length) * 8 ft (height) = 96 sq ft * 2 = 192 sq ft
- Top and Bottom faces: 12 ft (length) * 8 ft (width) = 96 sq ft * 2 = 192 sq ft
- Left and Right faces: 8 ft (width) * 8 ft (height) = 64 sq ft * 2 = 128 sq ft
Total surface area of rectangular prism = 192 + 192 + 128 = 512 sq ft
Surface area of right triangular prism:
- Front face: 8 ft (width) * 15 ft (hypotenuse) / 2 = 60 sq ft
- Left face: 8 ft (width) * height = 8h sq ft
- Right face (not visible): 8 ft (width) * height = 8h sq ft
We can use the Pythagorean theorem to find the height of the right triangular prism:
height = sqrt(hypotenuse^2 - base^2) = sqrt(15^2 - 8^2) = sqrt(225 - 64) = sqrt(161) ≈ 12.69 ft
Now we can find the total surface area of the right triangular prism:
Total surface area = 60 + 8(12.69) = 60 + 101.52 = 161.52 sq ft
Total surface area of the composite figure = surface area of rectangular prism + surface area of right triangular prism
= 512 + 161.52 = 673.52 sq ft
Therefore, the surface area of the figure is 673.52 square feet.