Problem 10 (The Pyramid of Oranges)

Here's a fun problem. Take 14 spheres, each one of radius 1, and stack them in the form of a pyramid (like they do in grocery stores when they stack oranges), putting 9 spheres at the base, 4 on top of those, and 1 at the very top as shown in Figure 1.

What portion of the volume of the "covering" pyramid is filled with these spheres ?

Solution

We know that the base of the covering pyramid is a square with corner points A, B, C, and D as shown in Figure 2 below.

If we can find the altitude and base of the triangle, we can find its volume and hence the solution of the problem. To find the altitude, we consider the triangle C1C3C9 shown in Figure 2. We conclude that C₁C₃ = C₃C₉ = 4 inches, and thus C₁C₉ = 4 sqrt (2). We now draw the vertical triangle shown in Figure 3.

and make the following conclusions

Hence, we have that the altitude of the pyramid is

Note: This proportion will approach 1 as the number of spheres increases. A good (but not easy) problem would be to determine this proportion as a function of the number of spheres.

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Last modified on Tuesday, January 12, 1999