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→Arc length and area: arc length and area relation |
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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.
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:
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