How to find the VOLUME of HEXAGONAL PRISMS?? PLease help!!!?

Question

Ok so here is the question:

How much plaster of paris is needed to make four miniature pillars for a model of a home if the pillars are regular hexagonal prisms with a height of 12 in. and face width of 2 in.

Oh and if you know, how can find the side of a square if you only have the diaganol length??

Thank you soooooooooooooooo much for your help

Answer 1

Volume of a prism is base area times height.

First you need to figure out the area of a regular hexagon with two-inch sides. The area of a regular polygon is the perimeter, times the apothem (distance from center of the shape to the center of any side), divided by two.

The apothem of a hexagon splits an equilateral triangle (from the center to two adjacent vertices), forming a 30:60:90 triangle, whose sides are in ratio 1:sqrt(3):2. In this case, 1=half of the side length, sqrt(3) is the apothem.

So...

area = perimeter * apothem / 2
area = (6 * 2) * (2 * sqrt(3) / 2) / 2
area = 12 * sqrt(3) / 2
area = 6 *sqrt(3)

Now you can solve the volume:

volume = area * height
volume = 6 * sqrt(3) * 12
volume = 72 * sqrt(3) cubic inches
volume = about 124.7 cubic inches

That's the volume of one pillar, the volume of four pillars is four times that much:

4 * volume = 4 * 124.7 = 499 cubic inches

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The diagonal of a square forms a right triangle with two of the sides. Since the two sides are the two legs of the right triangle, you can use the pythagorean theorem to solve it.

Let's call "s" the length of the side of the square, and "d" the length of the diagonal:

s^2 + s^2 = d^2
2 * s^2 = d^2
sqrt(2 * s^2) = sqrt(d^2)
sqrt(2) * s = d
d / sqrt(2) = s

The diagonal length, divided by the square root of 2, is the length of the side of the square.

Answer 2

Every time i ask a question, even if it is the simplest one, nobody can provide me a good informed answer on this site. Wtf happened to people who actually make the effort to answer..