To find the surface area of this prism, you need to add up the area of all the individual shapes outlined in your image.
Let's start from the top of the image.
The area of any given triangle is given by [tex] \frac{1}{2} [/tex]×base×height.
This 6-6-6 triangle is an equilateral triangle.
We have that the base is 6'.
Pythagoras rule gives us the height, [tex]h[/tex] as follows:
[tex]h^2=6^2-3^2=36-9=27[/tex]
[tex]h= \sqrt{27} =3 \sqrt{3} [/tex]
Thus, the area of the first triangle is:
[tex]A_1= \frac{1}{2}\times6\times3 \sqrt{3} =9 \sqrt{3} [/tex].
Now onto the middle section of the image.
The area of any rectangle is given by the length×width.
We have 3 equivalent rectangles such that finding the area of one of them will give us the areas of the other two.
The length is given as 8'; width is given by 6'.
Then, the area of one rectangle is 6×8 = 48.
Thus, the total area of all three rectangles is:
[tex]A_2 = 3\times48=144[/tex]
Finally, the bottom part of the image.
We have two identical right-angled triangles.
Calculating the area of one of them is sufficient.
Since base length is given by 3' and height is given by 5.2', the area is
[tex] \frac{1}{2} \times 3 \times 5.2=7.8[/tex]
Multiplying this by 2 gives us the total area of both triangles:
[tex]A_3=2\times 7.8=15.6[/tex]
Adding all three sections will give us the surface area of this prism:
[tex]A=A_1 + A_2 + A_3[/tex]
[tex]A=9 \sqrt{3} +144+15.6[/tex]
[tex]A=175.18845726812[/tex]
Therefore, the surface area is given by 175.18845726812.
