In H. G. Wells’s novel The First Men in the Moon, our natural satellite is found to be inhabited by intelligent insect creatures who live in caverns below the surface. These creatures, let us assume, have a unit of distance that we shall call a “lunar.” It was adopted because the moon’s surface area, if expressed in square lunars, exactly equals the moon’s volume in cubic lunars. The moon’s diameter is 2,160 miles. How many miles long is a lunar?
The volume of a sphere is 4π⁄3 times the cube of the radius. Its surface area is 4π times the square of the radius. If we express the moon’s radius in “lunars” and assume that its surface in square lunars equals its volume in cubic lunars, we can determine the length of the radius simply by equating the two formulas and solving for the value of the radius. Pi cancels out on both sides, and we find that the radius is 3 lunars. The moon’s radius is 1,080 miles, so a lunar must be 360 miles.
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A version of this puzzle originally appeared in the February 1960 issue of Scientific American.