Area of a CrescentDate: 11/10/2000 at 16:04:19 From: Dennis Graham Subject: Formula for area of a crescent I am an 8th grade math teacher. The elementary art teacher asked me for a formula for the area of a crescent. They were working on geometric shapes and discussing area. I have not found anything as of yet. Date: 11/10/2000 at 22:46:18 From: Doctor Peterson Subject: Re: Formula for area of a crescent Hi, Dennis. The answer, like any formula, will depend on what dimensions they know about the crescent. I'll suppose that they are just asking for a general formula, using whatever numbers make it easiest. But I can't think of any way to define it so as to give a really simple formula. Here's one approach: *********** C **** **oooooooooo *** ooo/ \ ***.....oooo ** oo / \ **.......oo * oo / \ *........oo * o /r1 \r2 *.........o * o / \ *.........o * o / \ *.........o *-------------o--+------------+---*E--------oF * o A\ d /B *....d....o * o \ / *.........o * o \ / *.........o * oo \ / *........oo ** oo \ / **.......oo *** ooo\ / ***.....oooo **** **oooooooooo *********** D The area of the shaded crescent, bounded by arcs of a circle with center A and radius r1, and another circle with center B and radius r2, can be found by adding quadrilateral ADBC and sector BDFC, then subtracting sector ADEC. You can find the sector areas if you know the angles DBC and DAC; to do that I would use triangle ABC and double the angles. The area of this triangle also gives you half the area of ADBC, and can be found using Heron's formula with r1, r2, and d. So it can be done, but I'm not interested in writing it out as a single formula. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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