finding the radius of two concentric circles with known arc length

Question

I am trying to figure out the radius length R to the center of the smaller of the two concentric circles. The length of the arc on the larger circle is 13.4 meters and the length of the arc on the smaller circle is 3.7855 meters. the distance between the circles is 3.048 meters I figured out generally that the radius is proportional to the ratio of the arc lengths, but the distance between the two circles plays into it and I haven't been able to figure out how. Thank you.

Answer 1

Let $\theta$ be the inscribed angle corresponding to the arc in radians, then we know:

$$R\theta=3.7855$$

$$(R+3.048) \theta=13.4$$

This is a system of two equations in two unknowns, which is easy to solve.

Answer 2

You’re most of the way there. $13.4 : 3.7855 :: R+3.048 : R$. Solve for $R$.

Answer 3

HINT

The similarity principle of arcs is no different from that of triangles contained between them.

$$ \frac{R}{R+L}= \frac{3.78}{13.4}$$

Using Componendo Dividendo Rules for fractions

$$ \frac{R}{L}= \frac{3.78}{9.62};$$