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.

Thus

.

Similarly for the case with *n* circles, we simply have a different value for

*θ*: .

Figure 2: The n=4 case

Thus,

.

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Tags: Circles, Geometry, Math, Monday Math, Trigonometry

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