Quote:
Originally Posted by Varyon
A standard hex is one yard wide and 4 yards tall. Assuming a circular base, it ends up being exactly pi cubic yards in volume.
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A hex does
not have a circular base, but a hexagonal one.
Assume a distance of 1 yard from hex center to hex center. Then the distance from hex center to the middle of a given edge is 0.5 yard. We can divide the hex into six equilateral triangles. Taking the line from the hex center to the middle of the edge as the adjacent side, and applying a bit of trigonometry, we get that the line from the center to the vertex is 0.57735 yards, and therefore any exterior side must be the same length, because the triangle is equilateral. Its area is therefore 0.1443 square yards, and the area of the entire hex is 0.866 square yards, or just about 13/15 square yards. Since the height is 4 yards, the volume is 3.464 cubic yards.
For comparison, an inscribed circle has radius 0.5 yards and area 0.785 square yards, and a circumscribed circle has radius 0.57735 yards and area 1.047 square yards. The former gives volume equal to pi; the latter gives volume equal to 4.1888 cubic yards.