I have a problem calculating the area of a circle segment.
I know how to separate this into smaller tasks (triangle and remaining circle segment) that are basically easily solvable, but one distance is missing and I have no clue how to determine it.
As you can see I need distance EB (which equals approx. 0.49597*r (by measuring)) to calculate the red area in the picture below:
From there the triangle would be easy and the remaining segment can be calculated by using a formula I found on wikipedia:
This might be a really simple problem but all the people I talked to could not come up with a solution so I am really hoping somebody here can figure this out.
Best Answer
You didn't give the radius. I assumed $2$.
The tangency point is at $(\sqrt2,\sqrt2)$, so that the center of the second circle is at $(2\sqrt2,2\sqrt2)$.
Its equation is $(x-2\sqrt2)^2+(y-2\sqrt2)^2=4$.
Intersecting with $y=2$ gives $x=2\sqrt2-\sqrt{4-(2-2\sqrt2)^2}\approx1.0080676825$ and $EB\approx0.9919323175$.
It will be easier to find the area by direct integration between $x_0$ and $x_1$ of the difference between the circle and the horizontal at $y_0=-\sqrt{4-x_0^2}$ (circle centered at the origin). $$A=\int_{x_0}^{x_1}(y_0+\sqrt{4-x^2})dx=(xy_0+\frac12\sqrt{4-x^2}+2\arcsin\frac x2)\Big|_{x_0}^{x_1}$$