Question 1659
To the nearest tenth find the volume of a regular hexagonal pyramid with base
edges of 12cm long and height of 15cm
`<font face = "courier new" color = "blue" size = 3><b>
Volume of a pyramid = Area of the base * Height * 1/3
`
So we begin by finding the area of the base.
The base is a regular hexagon.  A regular hexagon is 6 congruent
equilateral triangles placed together like 6 pieces of pie:
`__
<u>/\/\</u>
\<u>/\</u>/
`
The area of one equilateral triangle is given by A = bh/2. The base
is the edge and the height is found by the Pythagorean theorem
`
` `/|\
12/h| \12
`/_`|`_\
` 6 ` 6
`
6² + h² = 12²
36 + h² = 144
` ` `h² = 144 - 36
` ` `h² = 108
` ` ` {{{h = sqrt(108)}}}={{{sqrt(36*3) = 6sqrt(3)}}}
`
So the area of one equilateral triangle with edge 12 is
`
{{{A = bh/2}}}={{{12*6sqrt(3)/2 = 36sqrt(3)}}}
`
The area of the regular hexagon base of the pyramid is then
6 times this, or {{{216sqrt(3)}}}
`
Now since 
`
Volume of a pyramid = Area of the base * Height * 1/3, we have
`
Volume = {{{216sqrt(3)*15*(1/3)}}} or, {{{1080sqrt(3)}}}
`
Edwin <font face = "wingdings" size = 7 color = "red">J</font></b>