I have been wracking my brain on this problem and cannot derive the equation. It is related to tooth profiles on spur gears.

Given three equidistant concentric circles, with the inner circle radius r, the middle circle radius r+d, and the outer radius r+2d. A line intersects all three circles such that the angle formed by an intersecting radius-line with the middle circle is θ.

What are the resulting arc-angles α¹ and α² that result from the line intersecting the inner and outer circles? I actually need the sum of the angles, so if it is easier to solve as a single arc, even better. Thanks!

image describing the geometry


1 Answer 1


If I got your question right, you just need to apply sine theorem to the triangles formed by (1) radius $r+2d$ radius $r+d$ and the intersecting line, and (2) radius $r+d$ radius $r$ and the intersecting line. The angles of triangle (1) are $\alpha_2$, $\,\,\theta - \alpha_2$ and $\pi - \theta$ and (2) $\alpha_1$, $\,\,\theta$ and $\pi - (\alpha_1 + \theta)$. Hence $$\frac{r+2d}{\sin(\theta)} = \frac{r+d}{\sin(\theta - \alpha_2)}$$ $$\frac{r+d}{\sin(\alpha_1+\theta)} = \frac{r}{\sin(\theta)}$$ Solve for $\alpha_2$ and $\alpha_1$

  • $\begingroup$ But is θ the (top) angle of the triangle? Because of the radius angle, I think that top angle is less than θ (that is, the two lines are not parallel). $\endgroup$ Oct 9, 2016 at 4:57
  • $\begingroup$ Which one is theta? It is not very clear from your picture... $\endgroup$ Oct 9, 2016 at 5:08
  • $\begingroup$ @MarkWarren What about now? Is this what you mean? $\endgroup$ Oct 9, 2016 at 5:24
  • $\begingroup$ I think that is correct. What I wasn't seeing is that the top angle is θ - α², so knowing that, your solution makes sense. Thanks. $\endgroup$ Oct 9, 2016 at 14:08

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