Question

In a regular hexagonal prism, the longest diagonal is 12 ft. Find the volume of the prism if the diagonal makes an angle of 30 degrees with a lateral edge.

Answer 1

Because it's 'regular', I think that means the sides will be squares rather than rectangles.
So the length of each side of the hexagon will be the same as the height of the prism.
cos 30 = x/6 so x = (3) (sqrt3)
From the diagram above, the length of each side = 2 x (3)(sqrt3) = 6 (sqrt3) = 10.39 feet
So the height is approximately 10.39 feet.
There's a formula for the area of a hexagon but I'm just going to calculate it another way. The area of the hexagon equals the sum of area of the 6 triangles.