Consider the above picture, where you would want to find a value for $r$, depending on $R$ so that the conjoint area (marked gray) is equal to half the area of the $\color{darkorange}{\text{orange circle}}$, i.e.
$$A(r) = \frac{1}{2}\pi R^2$$
Come up with a way to find $r$, if the $\color{blue}{\text{blue circle's}}$ centre is located on the perimeter of the $\color{darkorange}{\text{orange circle}}$.
I'm okay with using Mathematica if necessary, I just can't seem to find a way to solve my problem. Also allow me to streamline the wording of the problem.
Problem worded differently
If you have a circle with radius $R$, and you attach another circle, with its centre, on the perimeter of the first circle. Find the radius $r$ (see first image) for the second circle such that the area between the circles is $50\%$ of the area of the first circle.
Best Answer
I don't use analytic geometry.
Let $A$ and $B$ be the points of intersection of the circles and $C$ the center of the circle of radius $r$.
Let $x=\angle ACB$ our unknown. Then $$r=2R \cos \frac x2$$ Simple facts about geometry and trigonometry give the equation $$\sin x - x \cos x= \frac \pi2$$ so $x$ is approximately $1.9056957$ and $r/R$ is approximately $1.1587285$ .