Question 891166: Hello, I'm having trouble with finding the volume of a regular hexagonal prism with measurements 4 and 9? Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! presumably the 4 is the length of a side of the hexagonal base and the 9 is the height of the prism?
the volume is equal to the area of the base * the height.
each interior angle of the hexagon is equal to (6-2) * 180 / 6 = 4 * 180 / 6 = 4 * 30 = 120 degrees.
the hexagon has 6 triangles whose sides have a vertex at the center of the hexagon and whose sides have the other end terminating in the 6 vertices of the hexagon.
this forms 6 triangles with 60 degrees all around which makes them equilateral triangles.
you need to find the area of one of these triangles and then multiply that by 6 to get the area of the base.
all sides of the triangle are equal to 4 because the triangle is an equilateral triangle.
take one of these triangles and drop a perpendicular from the center vertex to the opposite side and you form a right triangle with sides of 2, 2*sqrt(3), and 4
the side that is equal to 2 is half the side of the hexagon.
the side that is equal to 4 is the side of the triangle that connects between the center of the hexagon and the vertex of the hexagon.
this is also the hypotenuse of the right triangle.
the side that is 2 * sqrt(3) is the height of the triangle.
the area of each of the 6 triangles is equal to 1/2 * 2*sqrt(3) * 4 which is equal to 4*sqrt(3).
the area of the base of the hexagonal prism is equal to 6 * 4 * sqrt(3) which is equal to 24 * sqrt(3)
the volume of the hexagonal prism is equal to the height * the area of the base which is equal to 9 * 24 * sqrt(3) = 216 * sqrt(3) which is equal to 374.12 cubic units.
this is confirmed by using the following online calculator.
