Compute the areas of portions of a circle defined by a point

Question

Given a circle with center (cx,cy) and radius r, and a point inside the circle (px,py), draw the lines x=px and y=py to divide the circle into four sections. How can I compute the areas of these four sections?

Answer 1

Wlog, we can assume the circle is $C(O, r)$; where $O=(0, 0)$ and $r$ is the radius. We can further assume that $P(x_p, y_p)$ is a point in the 1st quadrant and $d(O, P) \le r$.

Thus, the lines $x = x_p$ and $y = y_p$ divide that quarter of circle into $4$ regions as shown.

It is not difficult to see that [(yellow)region] can be found.

By using “area of sector – the area of the corresponding central triangle”, the areas of the following can be found:-

$[A] = [(green + brown)]$; and $[B] = [(blue + brown)]$.

---

Then,

$[A \cup B] = [(green + brown + blue)] = 0.25[whole circle] – [(yellow)]$

$[(Brown)] = [A \cap B] = [A] + [B] – [A \cup B]$

Once $[(Brown)]$ is known, the other $3$ regions can be found.