Radius from an Arc and a ChordDate: 06/08/99 at 19:38:11 From: Jackie Castellano Subject: Geometry, arcs If I know the height of an arc from the midpoint on a chord, and the length of the chord, can I find the radius of the circle of which the arc is a part? I can't find a formula. Date: 06/09/99 at 12:11:16 From: Doctor Peterson Subject: Re: Geometry, arcs Hi, Jackie. You could work out this formula from what is in our FAQ on Circle Formulas: http://mathforum.org/dr.math/faq/formulas/faq.circle.html Look down the page for segments of circles, namely c = ... = 2 sqrt[h(2r-h)] You're given h and c, and need to find r, so you can solve this for r as r = (c^2/(4h) + h)/2 c^2 + 4 h^2 = ----------- 8h You can work out the formula this way: *********** ***** ***** **** /| **** ** / | ** * / | * ** r / | ** * / |c/2 * * / | * * / | * * / r-h | h * * *--------+------------* * \ | * * \ | * * \ | * * \ | * ** \ | ** * \ | * ** \ | ** **** \| **** ***** ***** *********** We have a right triangle, and Pythagoras gives: r^2 = (r-h)^2 + (c/2)^2 r^2 = r^2 - 2rh + h^2 + c^2/4 2rh = h^2 + c^2/4 h^2 + c^2/4 4h^2 + c^2 r = ----------- = ---------- 2h 8h I hope that meets your needs. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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