3 of 4 added 50 characters in body

How can I estimate or calculate the area of a circular segment?

Given a circle (for simplicity, $x^2 + y^2 = 1$) and a chord on this circle parallel to the $x$ axis $y = p - 0.5$ ($p \in [0,1]$ being the only parameter I control), how can I estimate the relationship between $p$ and the ratio between the circular segment determined by the circle and the chord and the point $(0,-1)$?

I do not require full precision (I only have coarse control over $p$, after all), but if no readily available approximation is available, I'll take the full formula.

In English — I have a circular shape in Powerpoint; I want to color a portion of its area with a given color. Powerpoint doesn't let me color a circular sector using the gradient fill tool, so I have to resort to a gradient section. The alternative is complicating the drawing with the arc tool and its fiddly controls. Thus, I'd like to know what portion of the circle I'd color if I color what's below a chord placed at $p$% height of the circle with red and the remainder in white.

enter image description here

What I tried

Wikipedia says that for this circle:

enter link description here

the area is: $$A = R^2 / 2 \cdot (\phi - \sin\phi)$$ where:

  • $R = 1$ (I'm only interested in the area ratio)
  • $\phi = 2 \arccos \frac{d}{R} = 2 \arccos d$
  • $d = 2(0.5 - p) = 1 - 2p$ for $p < 0.5$

So I'd have: $$A = \arccos(1 - 2p) - \sin \arccos (1 - 2p)$$

...which is absurd, because the area for $p = 0.5$ would be $\frac{\pi}{2} - 1$ instead of $\frac{\pi}{2}$.