Spheres inside a cone - Geometric mean

Question

If you have an odd number of balls (all touching each other) inside a cone so that they all touch the curved wall of the cone, why is the radius of the central ball the geometric mean of the radii of the balls at either end?

Answer 1

Here is the definitive answer: Let the smallest ball have radius $r_1$. Let the largest ball have radius $r_3$ and the middle ball have radius $r_2$. Then, by considering linear ratios (using the fact that the gradient of the tangent plane of the spheres coinciding with the cone is a constant given by $\tan\theta$, where $\theta$ is the angle between the tangent plane and the line through the centres of the spheres), we have $$\frac{r_2}{r_1}=\frac{r_3}{r_2},$$ giving $$r_2=\sqrt{r_1r_3}.$$ This is why we get the geometric mean.

Answer 2

Hints:

• the radii may be be $r, kr, k^2r$ using similar figures

• if so, the volumes are proportional to $r^3,k^3 r^3, k^6 r^3$

• note that $\sqrt{r^3 \times k^6 r^3} = k^3 r^3$