### Lunars

H.G. Wells's novel 'The First Men in the Moon' features intelligent insect-like beings who live in caverns below the surface.

Let us assume that these creatures have a unit of distance called a 'lunar'. It was adopted because the Moon's surface area, expressed in square lunars, exactly equals the Moon's volume, expressed in cubic lunars.

The Moon's diameter is 2160 miles.

How many miles long is one lunar?

(For the purposes of this puzzle, you may assume that the Moon is a perfect sphere).

## 2 Comments:

Let r be the radius of the Moon in miles = 2160/2 = 1080.

Let l be the ratio (1 lunar)/(1 mile).

Therefore (r*l) will be the radius of the Moon in lunars.

The surface area in square lunars = 4 * pi * (r*l) * (r*l)

The volume in cubic lunars = (4/3) * pi * (r*l) * (r*l) * (r*l)

As surface area in square lunars = volume in cubic lunars,

4 * pi * (r*l) * (r*l) = (4/3) * pi * (r*l) * (r*l) * (r*l)

With some cancelling out,

4 = (4/3) * r * l

3 = r * l

l = r/3 = 1080/3 = 360 miles.

One lunar is 360 miles.

You what else is one Lunar? Clue: it is accomplanied by two spheres.

JS

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