What is the surface area and volume of this triangular prism?

Question

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Answer 1

Volume is (triangle area) × length = (1/2)(6)(5)(12) = 180 ft^3.

The surface area is quite nasty. Using Pythagoras you can see that the part of the base to the left of the perpendicular is √(6^2 - 5^2) = √11, and hence the rest of the base has length 6 - √ 11. So the remaining side of the triangle has length
√(5^2 + (6 - √11)^2)
= √(25 + 36 - 12√11 + 11)
= √(72 - 12√11)
= 2√(18 - 3√11).

Now the area of the prism consists of the base (6×12 rectangle, area 72 ft^2), two copies of the triangle each with area 1/2 (6)(5) = 15 ft^2, the left-hand face (a 6×12 rectangle, area 72 ft^2) and the right-hand face (length 12, width 2√(18 - 3√11) and therefore area 24√(18 - 3√11)).

So the total area is 72 + 2(15) + 72 + 24√(18 - 3√11)
= 174 + 24√(18 - 3√11).
= 242.1 ft^2 to 1 decimal place.