Math CentralQuandaries & Queries


Question from David:

Im trying to form a hexagonal pyramid out of plywood and need to know the height to the center point of the pyramid. I want to use standard 4x8 sheets of plywood and split them diagonally. Then with the resulting triangles combine them to form one triangle. Then i want to accumulate 6 of these triangle total and put together to form a hexagonal pyramid.

Hi David.

I'll draw one of the triangles on a sketch and label some useful lines and points:

According to your plan, AD should be 4 ft and DT should be 8 ft. Therefore AB is also 8 ft.

The length CT is what you want to find.

All the way around a full circle is 360°. Therefore because of symmetry, angleBCA must be 360°/6 = 60°. Since AC = CB by symmetry, that means all the angles of triangleBCA are 60, so it is equilateral making AC also equal 8 ft.

If AC is 8 ft and AD is 4 ft, using the Pythagorean Theorem, CD2 = 82 - 42.
But if you look at triangleDCT, it is also a right triangle, so DT2 = CD2 + CT2. This means that
CT2 = DT2 - CD2. But that just means
CT2 = 82 - (82 - 42) = 42.

So CT, the height of the pyramid is 4 ft.

Hope this helps,
Stephen La Rocque.

PS: If you are interested in the angles to cut, then check this earlier question Chris and Harley answered.

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