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

The diagonal comes from the vertex on one base to the opposite vertex on the other base.

The lateral edge and the long diagonal therefore join a vertex one base to two opposite vertician the other and if we draw a diagonal between those vertices we will have a right triangle with angle if so and $60$ degrees with a hypotenuse measuring $12$ feet having the properties of side in such a right triangle we see that the lateral edges of prism measure.

$12$ ft $\left(\sqrt{3}\right)/2=6\sqrt{3}$ feet which is of where height of prism.

We above see that the long diagonal of a base (the shortest side of the right triangle) measure $=\frac{12}{2}$ feet $=6$ feet. Therefore sides of bases measure $3$ feet.

Sub diving the bases into $30-60-90$ triangles we get up then of base which is $\frac{\sqrt{3}}{2}$ times $3$ feet $=\frac{3\sqrt{3}}{2}$ feet.

Area of base $11$ half of a pothen and perimeter

$6\times 3$ feet $\times \frac{3\sqrt{3}}{2}$ feet $=27\sqrt{3}$ sq feet.

Multiplying that by height of prism yields.

Volume $\frac{2+\sqrt{3}squarefeet}{6\sqrt{3}feet}=486$ cubic feet.