Editing Circular segment (section)

=== Arc length and area ===
The arc length, from the familiar geometry of a circle, is
:<math>s = {\theta}R</math> 

The area ''a'' of the circular segment is equal to the area of the [[circular sector]] minus the area of the triangular portion (using the double angle formula to get an equation in terms of <math>\theta</math>):

:<math>a = \tfrac{R^2}{2} \left(\theta - \sin \theta\right)</math>

In terms of {{math|''R''}} and {{math|''h''}},

:<math>a = R^2\arccos\left(1-\frac{h}{R}\right) - \left(R-h\right)\sqrt{R^2-\left(R-h\right)^2}</math>

In terms of {{math|''c''}} and {{math|''h''}},

:<math>a = \left(\frac{c^2+4h^2}{8h}\right)^2\arccos\left(\frac{c^2-4h^2}{c^2+4h^2}\right) - \frac{c}{16h}(c^2-4h^2)</math>

What can be stated is that as the central angle gets smaller (or alternately the radius gets larger), the area ''a'' rapidly and asymptotically approaches <math>\tfrac{2}{3}c\cdot h</math>. If <math>\theta \ll 1</math>, <math>a = \tfrac{2}{3}c\cdot h</math> is a substantially good approximation.

If <math>c</math> is held constant, and the radius is allowed to vary, then we have<math display="block">\frac{\partial a}{\partial s} = R</math>

As the central angle approaches π, the area of the segment is converging to the area of a semicircle, <math>\tfrac{\pi R^2}{2}</math>, so a good approximation is a delta offset from the latter area: 

:<math>a\approx \tfrac{\pi R^2}{2}-(R+\tfrac{c}{2})(R-h)</math> for h>.75''R''

As an example, the area is one quarter the circle when ''θ'' ~ 2.31 radians (132.3°) corresponding to a height of ~59.6% and a chord length of ~183% of the radius.{{Clarify|date=December 2021|reason= A diagram with these numbers would be a good addition to the example}}