# How long is this line making a loop?

Here I have this loop, made of parts of two different circles with radiuses $$r_1$$ and $$r_2$$, joined with two lines intersecting at $$90$$ degrees and touching the circles only in one point, as shown in the picture below. How to calculate the length of a way from one point on the loop back to the same point?

• You need to decide how large the angle is. – Saucy O'Path Apr 7 at 13:14
• @SaucyO'Path Or, alternatively, how far apart the centers are. – Ethan Bolker Apr 7 at 13:16
• @EthanBolker Yes; in fact I was about to write it. – Saucy O'Path Apr 7 at 13:16
• @SaucyO'Path 90 degrees, forgot to write that! ty in advance – Xxx Ddd Apr 7 at 13:21
• @EthanBolker 90 degrees, forgot to write that! ty in advance – Xxx Ddd Apr 7 at 13:22

With a right angle in the middle, connecting the centers of the circles to the points of tangency produces two squares. Then you can see that the distance $$s$$ all around the figure eight is $$3/4$$ of each circle plus twice the sum of the radii: $$s = \left(\frac{3 \pi}{2} + 2 \right)(r_1+r_2) = C(r_1+r_2).$$ If you know $$s$$ and $$r_1$$ then $$r_2 = \frac{s - Cr_1}{Cr_2}.$$
Note how that requires $$s > Cr_1$$. When $$s = Cr_1$$ the second circle is a point and there's a right angle corner in the figure.
• so if the length $s$ equals 280 and radius of the bigger circle $r_2$ is 25, how would you calculate the radius of the smaller circle? Like this - $r_1 = s / (3/2 \pi + 1) - r_2$ ? thanks for answering – Xxx Ddd Apr 7 at 15:12
• oh, stupid mistake, I added the $(r_1 + r_2)$ just once. – Xxx Ddd Apr 7 at 15:35