I'm trying to determine the perimeter of a teardrop shape formed by two adjacent circles (non-intersecting) with mutually tangent lines drawn on both sides of the circles. I've attached a sample diagram.
Known variables:
$$ \begin{array}{r|l} b = r_1 & \text{radius of the larger circle}\\ r_2 & \text{radius of the smaller circle}\\ t_1 & \text{tangent line between points p2 and p3}\\ t_2 &\text{tangent line between points p1 and p4}\\ \end{array} $$
I'm calculating the tangent line length by the following formula: length of tangent line $=2 \sqrt{r_1r_2}$
I've also assumed that $t_1 = t_2$. Is that a valid assumption?
Next, I would need to calculate the length of arc1 and arc2 (Arc1 is the arc between points p3 and p4, and arc2 is the arc between points p1 and p2). To do this, I believe I have to calculate the angle adjacent to the hypotenuse. I used the following:
$$p = \arcsin\left(\frac{r2-r1}{t1}\right) \frac{180}{\pi} $$
Then, I used the following to get both arcs. :
$$\begin{align} arc_1 = 2 r_1 \pi \frac{180-2p}{360}\\ arc_2 = 2 r_2 \pi \frac{180+2p}{360} \end{align}$$
Finally, total perimeter would be:
Perimeter of shape $= 2 t_1 + arc_1 + arc_2$
Did I do that correctly? Did I miss something? I think it's close just based on drawing and measuring, but I would like to get the formulas correct.
