Results 1 to 3 of 3

Math Help - Finding the area of a circle segment with very limited information.

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    2

    Finding the area of a circle segment with very limited information.

    I am trying to help my grandfather who is a making a model sail boat. The sails are isosceles triangles with a curve along the largest edge. He needs the area in square inches of the sails. I have found the area of the triangles themselves, but i cant seem to come up with a method of getting the area of the curved part, which is basically a circle segment. The only information i have on them is the chords and the distance between the chord and the arc.

    Length of chord: 54 in.
    Distance to arc: 2 in.

    If i can see the method i can use it on the other sail. But after searching various data banks i can not find anything. I have found the area using radius of the circle and degree of the angle made from the two points of the chord, but i do not have either.

    Any help would be much appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,711
    Thanks
    630
    (Sleepy)Hello, Gedalas!

    I am trying to help my grandfather who is a making a model sailboat.
    The sails are isosceles triangles with a curve along the largest edge.
    He needs the area in square inches of the sails.
    I have found the area of the triangles themselves, but i can't seem to come up with
    a method of getting the area of the curved part, which is basically a circle segment.

    The only information i have on them is the length of the chord
    . . and the distance between the chord and the arc.

    . . Length of chord: 54 in.
    . . Distance to arc: 2 in.

    I will assume that the circular arc is outside the isosceles triangle.


    Code:
                        C
                     *  *  *
                *       |2      *
            *           |           *
        A * - - - - - - + - - - - - - * B
           \     27     |D    27     /
            \           |           /
             \          |          /
              \         |         /
               \        |        /
              R \       |R-2    / R
                 \      |      /
                  \     |     /
                   \    |    /
                    \   |   /
                     \  |  /
                      \ | /
                       \|/
                        *
                        O

    We are given: . AB \:=\:54,\;\;CD = 2

    Let the radius be R.
    . . Then: . OA = OB = OC = R
    . . And: . OD = R - 2


    In right triangle BDO:\;\;(R-2)^2 + 27^2 \:=\:R^2

    . . R^2-4R + 4 + 729 \;=\;R^2 \quad\Rightarrow\quad 4R \:=\:733

    Hence, the radius is: . R \;=\;\dfrac{733}{4} \;=\;183.25\text{ in}


    In \Delta AOB, apply the Law of Cosines:

    . . \cos(\angle AOB) \;=\;\dfrac{183.25^2 + 183.25^2 - 54^2}{2(183.25)^2} \;=\;0.956582026

    . . \angle AOB \:=\:16.9455792 \;\approx\;17^o



    The area of sector ACBO is:

    . . \dfrac{17^o}{360^o}\pi(183.25)^2 \;\approx\;4981.78\text{ in}^2 \;\approx\;34.60\text{ ft}^2

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2010
    Posts
    2
    First off, my apologies on saying isosceles, it was a slip of the fingers, it is an acute triangle, but that does not affect your solution which does indeed help me. Thank you! You gave me the area of the sector, while i needed the area of the segment, anyway, i was able to use your radius and interior angle answers and plugged them into the A= (R^2/2)(theta-sin(theta)) for the circle segment. If anyone is interested the answer is approx. 72.77 in. sq.
    Last edited by Gedalas; August 29th 2010 at 05:45 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subtracting Vectors - Limited Information
    Posted in the Trigonometry Forum
    Replies: 8
    Last Post: March 25th 2011, 06:33 AM
  2. Replies: 3
    Last Post: July 23rd 2010, 04:19 PM
  3. Polar equations, limited information given
    Posted in the Pre-Calculus Forum
    Replies: 8
    Last Post: May 23rd 2010, 05:12 PM
  4. Replies: 2
    Last Post: October 10th 2009, 07:56 AM
  5. area of a segment of a circle
    Posted in the Geometry Forum
    Replies: 4
    Last Post: February 12th 2009, 11:44 PM

Search Tags


/mathhelpforum @mathhelpforum