Topical Guide 352
Mark Dorset
Sand
Mathematics (Applied): Estimating Number of Grains of Sand in a Given Container or Location
“You know,” Edwin interrupted, “a few months ago I built my daughter a sandbox. I had to figure out how much sand to get for it. Simple problem. Nothing to it. But a question popped into my head: how many grains of sand do I need? I made myself a little cardboard cube a quarter inch on a side, filled it with sand, poured the sand out on a sheet of paper and — ”
“ — counted the grains?”
“ — counted the grains. When I finished, I did it again — ”
“ — with a different sample of sand,” suggested Lorna.
“ — with a different sample of sand,” said Edwin …Herb ’n’ Lorna, Chapter 14
How many grains of sand are there in the world? Who better to work out the answer to 10-year-old Stanley’s question than one of Britain’s leading mathematicians?
Mathematician Marcus du Sautoy replies: Sand is rock with a diameter of between 0.625mm and 0.2mm. Assuming a grain of sand is roughly spherical, the average volume of a grain is 4/3 x pi x r^3 = 0.00947mm^3, where r is the radius. So how many grains of sand are there in a metre cube box? It has 10^9mm cubes inside (10^N is how mathematicians write a 1 followed by N zeros), and if they are arranged randomly, about 65% of the box will be sand and the rest air. So we can estimate that the number of grains of sand in a metre cube box is 10^9 x 0.65/0.00947, or roughly 70bn grains. Now, let’s go for an average of 5% of the surface of the Earth being covered in sand with a depth of 100m. The surface area of the Earth is 4 x pi x r^2 where r is the radius of the Earth, which is 6,378,000 metres. So the volume of sand comes out at: 2.5 x 1015m^3. So my rough estimate is that the number of grains of sand on the Earth is a number with 27 digits.
The Guardian, “Ask a grown-up: how many grains of sand are there in the world?”
Note to self: Include “Sand” in The Topical Autobiography of Mark Dorset.
See also: Dust TG 79
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