Circular segment: Difference between revisions

Revision as of 17:32, 17 April 2022

Revision as of 06:15, 18 April 2022

Line 37:

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

Unfortunately, <math>a</math> is a [[transcendental function]] of <math>c</math> and <math>h</math> so no algebraic formula in terms of these can be stated. But 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.

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: