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 Question from Rebecca: I'm doing a project for my Honors Geometry class and I have to build a regular hexagonal pyramid. I was given the volume of .75 gallons, but I don't know how to figure everything else out. Please help?

Hi Rebecca,

Since you are in the US I think you probably want the dimensions of the pyramid to be in inches. I don't remember how many cubic inches are in a gallon and since I'm a macintosh user I asked Siri. She told me there are 231 cubic inches in a gallon so you want to build a pyramid of volume $0.75 \times 231$ cubic inches.

The volume of a pyramid is given by

$\mbox{ volume } = \frac13 \times \mbox{ the area of the base } \times \mbox{ the height}$

where the height is the vertical distance from the plane containing the base to the apex of the pyramid.

For the area of the hexagonal base I would look at Stephen's response to a question from Scott.

Use Stephen's construction to find an equation for the area of a regular hexagon in terms of the side length $a,$ and multiply by the height $h$ to get an equation for the volume. Use your calculator and some experimentation to find values of $a$ and $h$ to give the volume you need.

Good luck with this,
Harley

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