Circular segment: Difference between revisions

Revision as of 14:53, 14 January 2023

Revision as of 01:51, 14 February 2023

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:<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''}},

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.

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