Grains of Sand

Have you ever sat on a beach and wondered how many grains of sand there were?  I have, but I may be a special case.  Today we're going to take that a step further, and figure out how many grains of sand there are on the entire earth.  (Caveat: I'm only going to consider sand above the water level, since I don't have any idea what the composition of the ocean floor is).

I'm going to start by figuring out how much beach there is in the world.  If you look at a map of the world, there are four main coasts that run, essentially, a half circumference of the world.  We'll say the total length of coast the world has is roughly two circumferences.  As an order of magnitude, I would say that the average beach width is 100 m, and the average depth is 10 m.  This gives a total beach volume of

\[ (100 m)(10 m)(4 \pi (6500 km) )= 82 km^3\]

That's not a whole lot of volume.  Let's think about deserts.  The Sahara desert is by far the largest sandy desert in the world.  Just as a guess, we'll assume that the rest of the sandy deserts amount to 20% (arbitrary number picked staring at a map) as much area as the Sahara.  According to wikipedia the area of the Sahara is 9.4 million km^2.  We'll take, to an order of magnitude, that the sand is 100 m deep.  10 m seems to little, and 1 km too much.  That amounts to ~1 million km^3 of sand.

We're going to assume that a grain of sand is about 1 mm in radius  The volume occupied by a grain of sand is then 1 mm^3.  Putting that together with our previous number for the occupied volume gives

\[ \frac{1\cdot 10^6 km^3}{1 mm^3}=\frac{1 \cdot 10^{15}}{1\cdot 10^{-9}}=1\cdot 10^{24}\]

That's a lot of grains of sand.



Addendum:
Carl Sagan is quoted as saying

"The total number of stars in the Universe is larger than all the grains of sand on all the beaches of the planet Earth"

If we just use our beach volume, that gives a total number of grains of sand as ~1*10^20, which is large, but not as large as what we found above.  Is that less than the number of stars in in the universe?  Well, that's a question for another day (or google), but the answer is, to our best estimate/count, yes.