How to find the area of a semicircle inside of another semicircle?

Question

I am stumped on how to solve this problem. I've tried to look up the problem, but to no avail. What I need to know is how can I find the area of a semicircle cut off by the curve of another semicircle?

I have an example image here

The radius of the larger semicircle is 1 inch, while the radius of the smaller semicircle is 0.5 inches. I can easily understand how to find the area of the larger semicircle, but I cannot find the area of the smaller one, as it is not a true semicircle (the bottom is cut off by the curve of the larger semicircle.)

Any help or an answer would be appreciated, thanks.

Answer 1

The area of intersection between two circles is:

where $d$ is the distance between two centers, R and r are the two radii. In your case: $R = 1 in$, $r=0.5 in$ and $d=R=1 in$.

[source] http://mathworld.wolfram.com/Circle-CircleIntersection.html

Answer 2

N.B. $\operatorname{cs}(ACD)$ is the circle segment defined by the rays, $AC$ and $AD$.

In the piture below, we have that $$\operatorname{cs}(ACD)+\operatorname{cs}(BCD)$$ contains the part were looking for. However we have overcounted. Can you see by how much we overcounted (i.e. what we now need to substract)?