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.

Question

anonymous · Answer

The longest diagonal is the one that goes from a vertex of the top hexagonal base to the diametric opposite vertex of the bottom hexagonal base. Since that diagonal makes a 30° angle the the lateral edge (same as the height in this case), then it makes a 60° angle with the base itself. There is a 30°-60°-90° triangle in the interior of the prism which has 12 ft. length as its hypotenuse. That makes the height of the prism 6√3 ft., and the regular hexagonal base with a longest diagonal (from a vertex to its opposite vertex) of 6 ft. The that diagonal is 2 radii of the of the regular hexagon, making the sides of the hexagon 3 ft. A regular hexagon is made up of 6 equilateral triangles. The area of one equilateral triangle is s²√3/4 = 9√3/4, so the hexagon's area is 6·9√3/4 = 27√3/2 ft². Base area times height is volume of a prism = (27√3/2)(6√3) = 243 ft³.

anonymous · Answer

Thank You

anonymous · Answer

Np

anonymous · Answer

how did you get the 6 square root of 3?

anonymous · Answer

In a 30-60-90 triangle, the sides are in the ratio 1, √3, 2. Since 12 is the hyp, then the height is (12 times √3)/2=6√3

anonymous · Answer

Pls. help me to solve this. Thanks