Finding the surface area of a greenhouse I was given the following problem, which I feel should be simple but I seem to be getting the wrong answer:

Determine the surface area of the greenhouse with the dimensions below. Round to the nearest whole number.  Dimensions: 96' long, 23' wide, 12' high, 6'4" height from floor until roof.

This is my work: The greenhouse is a right triangular prism on top of a rectangular prism.
Surface area of rectangular prism:  2(wl+hl+hw) therefore: 2(2396+236.33+966.33)=5922.54
Surface area of right triangular prism: Need to find length of sides of triangle. $a^2+a^2={23}^2\\
 a^2=132.25$ and a=11.5
11.596=1104 – the area of one rectangular side. Need two.
23*5.67=130.41 – area of triangular side. Need two
Surface area of right triangular prism: 1104+1104+130.41+130.41=2468.82
2468.82+5922.54=8391.36
Solution: 8391’ 4”
Like I said, I don't think this is right, but I don't know where I went wrong. Any enlightenment?
 A: I believe the issue in your solution is that you computate $a^2 = 132.25$. If we check that you get: $$132.25 + 132.25 = 23^2$$ Or...
$$264.5 = 529$$
Which is clearly false.
The actual answer is that $a^2 = 264$ which means a is approximately $16.263455967$.

(The height of the pink area, d, is calculated using the total height of the structure, 12 ft, minus the height of the walls which was 6'4" or 6.33... ft)
Finding the surface area of A, the rectangular prism's length side, is as easy as $96*6.33...$ which equals $608$. Multiply that by two to account for the other side and get: $$1216$$.
The surface area of B, the rectangular prism's width side,  will also be simple as it is $23 * 6.33...$ which equals $145.66...$ . Again multiply by two to account for the other side resulting in: $$291.3333...$$ .
To find the surface of C, the triangular prism's side, we will need to multiply the length, 96, by the height of the side. The height of the side will be equal to a in the formula $a^2 + a^2 = 23^2$. As mentioned above, a comes out to be around 16.2480768. $96 * 16.263455967 = 1561.291772832$ . As always, multiply by two to account for the other side to get: $$3122.583545664$$.
Lastly, to find the surface area of D, the triangular prism's triangular side, we multiply ${\frac{23 * 5.66...}{2}}$ which equals $65.166...$ . Multiply by two for the last time to account for the other side to wrap up in: $$130.3333...$$
Add all these four values up of A, B, C, and D to finally get the total of:$$4629.916879 ft^2$$
as the surface area of the greenhouse.
A: While there are a few ways to go about solving this problem, there are surely a few mistakes in the work shown here, but they can be quickly solved: We can assume that because it is a triangular prism on top of a rectangular prism that the roof will not be considered in the "outside" surface area of the Greenhouse (I'm following the idea that we will later use this surface area relative to the sunlight coming in through this amount of surface area, therefore only considering the outside surface area). With this same logic, the surface area of the floor should not be considered either. Here is what we must solve for in this problem now:

*

*A (2) Long sides perpendicular to ground

*B (2) Short sides perpendicular to ground

*C (2) Triangle sides of the triangular prism

*D (2) Slanted ceiling surface rectangles

Please note that you did not specify in the original problem presentation the orientation of the triangular prism on top of the rectangular prism, so instead of assuming the triangles would be placed above the short sides of the green house, they could be above the longer sides, instead. (If you run into further issues, assumptions about dimensions may be withholding the correct answer)
Moving on from this, let's solve for the four parts of this figure:
A - Long sides perpendicular to ground
Length: 96', Height: 6 1/3', Area = 96*(19/3) = 32*19 = 608 * 2 (two sides) = 1216 ft^2
B - Short sides perpendicular to ground
Length: 23', Height, 6 1/3', Area = 23*(19/3) = 437/3 = 145.667 * 2 (two sides) = 291.333 ft^2
C - Triangle sides of the triangular prism (assuming above the short side of the rectangle, as proposed in the OP)
Please Note: In the OP, you assumed that the triangle was a right triangle, then using the Gougu (Pythagorean) Theorem, not the Law of Cosines, to find the length of the legs of the isosceles triangle. This triangle is actually not a right triangle:
Height (altitude from vertex adjoined to equal side lengths) = 12' - 6 1/3' = 5 2/3' . This is perpendicular to the long side of the triangle, being 23' under the OP's assumption; therefore, the area of the triangle is just (bh)/2  (not bh as shown in the OP) . Calculating this area is simply just: 23' x 5 2/3' = (23 x 17)/(3*2) * 2 (two sides) = 130.333 ft^2
D - Slanted ceiling surface rectangles
The length of these rectangles is 96' (as assumed in the OP), and we can find the height (triangle's shorter leg length) using the Gougu Theorem:
The base of the isosceles triangle is 23', and the height is 5 2/3'. Divide the base by 2 to find the leg of the desired right triangle (with the slanted ceiling's side length being the hypotenuse of the aforementioned right triangle). There is now a right triangle with base of 11 1/2' and height of 5 2/3'. Using the Gougu theorem, we get 11.5^2 + 5.667^2 = h^2, h = sqrt(132.25 + 32.111) = 12.8203 ft.
Now the length of the rectangle, 96', times the height, 12.8203', is 1230.7526 * 2 (two sides) = 2461.5 ft^2
Final Answer! - Summing all of these parts together, we get A + B + C + D = 4099
Thanks for the fun problem, and I hope that this helps! Go follow my Youtube Channel as I will be updating it in the future.
