Suppose we have three congruent, mutually (externally) tangent circles of radius r. If we circumscribe them with a larger circle of radius R, so that they are all internally tangent to the large circle, what is R in terms of r? What if, instead of three smaller circles, we have a symmetric ring of n≥2 circles inscribed inside the larger circle?
Let point O be the center of the large circle, point A be the center of one of the smaller circles, and B be one of that circle’s points of tangency with one of the other smaller circles. Let us denote the angle ∠AOB as θ.
Figure 1: Our construction
From our figure of this, we see that the length of segment OA is R–r, and that the length of AB is r. But the ratio of AB to OA is the sine of the angle θ. Here, we see θ is equal to π/3.
Similarly for the case with n circles, we simply have a different value for
Figure 2: The n=4 case