SOLUTION: What is the volume of a hexagonal pyramid when the base's side length is 3 and the angle from a point B at a vertex on the base, to point C at the radius, to point A at the top of

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Question 252828: What is the volume of a hexagonal pyramid when the base's side length is 3 and the angle from a point B at a vertex on the base, to point C at the radius, to point A at the top of the pyramid, is 35 degrees? Thank you SO much for your help.
Answer by drk(1908) (Show Source):
You can put this solution on YOUR website!
The first thing you notice is that we have a hexagonal based pyramid.
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step 1: find the area of the hexagon with side = 3.
These are pretty easy to find. Draw a segment from the center (C) to the vertex (B) and its adjacent vertex. This should make a triangle. Now each interior angle of a hexagon is 120 degrees, so our triangle is a
60-60-60. We use the formula for equilateral triangles,
(i)
where s = 3 to get
(ii) = 9sqrt(3)/4.
This is just 1 of the six triangles in the base, so we multiply (ii) by 6 to get
(iii) 54sqrt(3)/4 or reduced 27sqrt(3)/2.
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step 2 find the height of the pyramid. We know a right triangle with one angle = 35 degrees. If we drew the picture, BC = 3, angle B = 35, angle BCA = 90.
We can use
tan(35) = H/3 to get the height.
H ~ 3*.70021 ~ 2.10063 ~ 2.1
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step 3: Now we can find the volume of the pyramid using step 1 and 2 answers:

V ~ 16.37.