I struggle doing a Khan Academy exercise on density and area problems. One of them that I really do not understand is this:

Densities of Metal, a world-famous rock band, put on a show at an arena in Rome last night. The rectangular floor of the arena had an area of 32,000 meters squared and was packed shoulder to shoulder with fans. The band wants to know how many people were at last night's show. They estimate that, on average, each person takes up a circular space with a diameter of 0.8. Since the floor was packed, they assume these circular spaces were tightly packed together in rows and columns. Estimate how many people were at Densities of Metal's rock show based on their assumptions.

When I checked the answer to the problem, it said that I had to find the area of the square surrounding the circle ( this is a visual )https://cdn.kastatic.org/ka-perseus-graphie/71b810792d30f4f02d87f111a0fe89a07f29ffd5.png), and divide the area of the concert with that. What I don't understand is why you need to divide by the area of the square instead of the circle.


You can't fill a $32,000$ square meter arena with circles, and actually take up all of the space. Circles don't fit together like that; there's space left in between. If you circumscribe squares around the circles, however, you can pack the entire space with squares, and actually account for the area. The idea, I guess, is that the people are more-or-less in rows and columns.

This gets at an issue which isn't mentioned explicitly in the problem: circle packing. A square-type packing is one way to fill a space with circles pretty efficiently, but a triangular packing works better. I don't see, in this problem, why we would assume concert-goers approximate a square packing instead of a hexagonal one, except maybe ease of calculation.

See this article for more info: Circle packing


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