# Geometric expressions for a quarter circle cut at an arbitrary point

## Homework Statement

I am after finding general geometric expressions for a quarter-circle that is split into two segments along either its domain or range (they are equal). I.e. Taking the circle shown in Figure 1 and concentrating on the upper right quadrant, I am after expressions for the individual areas (top and bottom) and their respective centroids x1, y1 and x2, y2 (Figure 2) when cut at "c".

Figure 1:

Figure 2:

## Homework Equations

Area of circle = ∏r$^{2}$

Equation of quarter circle: y(x) = √(x$^{2}$-r$^{2}$)

## The Attempt at a Solution

Apart from stating the obvious equations (above) I’m stuck on this problem.

I have searched my textbooks/google (using general key words) for expressions of this particular case with no luck.

I have never been strong with deriving expressions from first principals and would appreciate any ideas/push in the right direction. This is a problem directed at self-study, not homework.

## Answers and Replies

If you immagine the other half of the circle you can use the equation:
$\bar{y}$=$\frac{4Rsin^{3}(\frac{1}{2}θ[STRIKE][/STRIKE])}{3(θ-sin(θ))}$
where θ is the theoretical angle to create the chord to get your y bar for the top section. Then you can calculate the y bar for the whole quadrant by using the equation:
$\bar{y}$=$\frac{4R}{3π}$
Since the average of the top and bottom y bar have to equal the quadrant based on their area you can take the weighted average of the top area and the y bar with the weighted average of the bottom area equaling the total area and the total y bar. From that you can get your y bars and then you can do the same thing for the x bars. Might sound confusing so if you don't understand let me know, I'll try my best.

Area of top part:
A=$\frac{R^{2}}{2}$(θ-sin(θ))