June 27, 2014
Our solution calculates the surface area of the planet then multiplies by the number of raindrops that fall at an average spot on the face of the planet.
Archimedes figured out long ago that the surface area of a sphere is 4 * pi * r^2. The earth has a circumference of 25,000 miles, or a diameter of 8000 miles, or a radius of 4000 miles, which is 12 * 5280 * 4000 = 250 * 10^6 = 1/4 * 10^9 inches. Thus the surface area of the Earth is 4 * pi * 1/4 * 1/4 * 10^18 square inches, the four constants (roughly) cancel each other, and the area is 10^18 square inches. Wolfram|Alpha agrees with our rough calculations; the exact answer is 7.9 * 10^17 square inches.
Figuring out the number of raindrops that fall per square inch is harder. We’ll assume that a raindrop is a cube 1/8 of an inch on a side, so there are 512 = 1/2 * 10^3 raindrops in a cubic inch. Some spots on the planet receive no rain (deserts) while other spots on the planet receive upwards of 120 inches (rain forest) of rain per year; we’ll arbitrarily say the average spot on Earth receives 20 inches of rain per year, which if it’s not right is at least within a factor of 2 or 3 from the right answer. Thus, each square inch of the Earth’s surface receives 1/2 * 10^3 * 20 = 10^4 raindrops per year, and the entire surface of the Earth receives 10^18 * 10^4 = 10^22 raindrops per year.
After solving the problem, I asked Google for other solutions. Oldguy makes similar calculations and comes to 7 * 10^18 raindrops per day, which is 2.5 * 10^20 raindrops per year, about 1/40th as much as our estimate. Doctor Ian estimates 10^19 cubic inches of rain fall per year, but counts only 16 raindrops per cubic inch, for a total of 1.6 * 10^20 raindrops per year, but his raindrops are much larger than ours, which fit 512 per cubic inch; if you make his raindrops and ours the same size, we would be within a factor of 2 of each other, which is comfortably close for this kind of estimate.
So the question comes down to how big is a raindrop, and how much is the average annual rainfall at any given spot on the planet.
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