Finding length of Arc knowing only Chord and Angle
Category: Science > Math
Asked by: joshuac-ga
List Price: $40.00
27 Sep 2006 22:19 PDT
Expires: 27 Oct 2006 22:19 PDT
Question ID: 769137
This may be embarrassingly simple, but I haven't been finding what I need out there: I would like a formula I can use with simple tools (pocket calculator) to calculate the length of an Arc knowing the length of the Chord and the Central Angle. (I'm using these terms as defined on the page http://www.1728.com/circsect.htm) For example, if I know my Chord is 50 units and that my Central Angle is 90 degrees (25% of a circle), with this information I would like to be able to directly calculate the Arc of ~55.54 units. If I could also find the Radius of ~35.36 units this would be helpful as well to me, but not necessary to answering the question. Information/tools I have found useful so far in my search: "Circle Calculator", http://www.1728.com/circsect.htm "Chord (geometry) - Wikipedia", http://en.wikipedia.org/wiki/Chord_%28geometry%29 I want to be able to do this calculation in the field, the simpler the tools needed the better. Acceptable answers, in order of best to worst: -Can be solved with pen and paper using simple math -Can be solved with scientific calculator (TI-36x manual; http://www.radioshack.com/sm-ti-36x-solar-scientific-calculator--pi-2104642_tb-support.html) -Can be solved using a Spreadsheet (MS Excel, OpenOffice.org Calc) Sorry my question is so wordy, I'm trying to define something I don't understand as well as I want to. If I left anything out you need to know please ask.
Re: Finding length of Arc knowing only Chord and Angle
Answered By: justaskscott-ga on 28 Sep 2006 01:02 PDT
Hello joshuac, Here's a page that sets forth the answer. "Ask Dr. Math: FAQ: Segments of Circles" The Math Forum @ Drexel http://mathforum.org/dr.math/faq/faq.circle.segment.html The top of the page provides these definitions: s = "length of the arc" c = "length of the chord" r = "radius of the circle" theta = "measure in radians of the central angle subtending the arc ((where 0 <= theta <= pi)" (1 degree = pi/180 radians) Further down the page ( http://mathforum.org/dr.math/faq/faq.circle.segment.html#9 ), you'll find the formulas: r = c/(2 sin[theta/2]), s = r theta Suppose, for example, you know that the chord (c) is 30 and the central angle subtending the arc is 90 degrees (90 * pi/180 radians, or approximately 1.571 radians). The radius would be 30 / (2 * sin[1.571/2 radians], or 30 / (2 * sin[90/2 degrees]). The sine of 1.571/2 (i.e., 0.7855) radians or 90/2 (i.e., 45) degrees is approximately 0.707. 2 times 0.707 is 1.414. 30 / 1.414 is approximately 21.22. Thus, the radius would be approximately 21.22. The arc (s) is r theta. In this case, it's approximately 21.22 * 1.571, or approximately 33.36. The calculation will be more precise if you don't round off these figures as much as I have in this example. The Math Forum page links to an Excel file ( http://mathforum.org/dr.math/gifs/ChordMath.xls ) which, among other things, automatically calculates of arc length after inputting chord length and theta in radians. A scientific calculator should be sufficient for performing these calculations. If you're doing the calculation on the Web, you may find these tools helpful: "Decimal Degrees And Radians Calculator" CSGNetwork and Computer Support Group http://www.csgnetwork.com/degradcalc.html "Virtual Calc98" [which can be used to calculate the sine of a particular number of degrees] The Calculator Home Page http://www.calculator.org/jcalc98.html - justaskscott Search strategy: Searched on Google for these terms in various combinations: arc chord length "arc length" "chord length" "inscribed angle" "central angle" circumference radius circle radians degrees calculator online
rated this answer:
and gave an additional tip of:
Beautiful, I even had gone through the mathforum.org website in my search and passed right over this (probably in part due to a limited geometry knowledge). Thank you for your detailed explanation as well.
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