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Area of a sector without knowing the angle (can't use a calculator)

  1. Jun 20, 2009 #1
    1. The problem statement, all variables and given/known data

    http://img10.imageshack.us/img10/9651/mathhelp.jpg [Broken]
    RED was given, BLUE was what I was able to find.

    What is the area of the shaded part?

    2. Relevant equations

    Area of shaded part = area of two triangles + area of sector

    3. The attempt at a solution

    I was able to find the area of the two triangles easily enough, but I can't think of a way to find the the value of angle AOB without using a calculator. My idea was to find angles AOP and BOQ and subtract them from 180, but again, no calculators allowed.
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jun 20, 2009 #2


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    Take a look at the formula for a chord segment.
    You don't need a calculator to find sin(x) if you know the lengths of the sides
  4. Jun 20, 2009 #3
    NumberedEquation6.gif (only sane looking formula I found)

    Is that it? To clarify, I'm trying to find the angle in order to find the sector area. Are you saying that there is a way to find the area without knowing the angle?

    I haven't studied this in quite some years.
    Last edited by a moderator: Apr 15, 2017
  5. Jun 20, 2009 #4
    The problem implicitly gives you the lengths of all three sides of a triangle. Can you think of a formula that will give you the angles within a triangle given all three sides?
  6. Jun 20, 2009 #5


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    Draw a line from A-B, there is a formula for the curve+chord shape which involves sin(theta) You can get sin theta from the sides of the triangle

    ps. I don't know if this is the answer - it's just something that occurs
  7. Jun 21, 2009 #6
    Once again, I fail to notice the simplest things. Triangle APO is a standard 30-60-90 triangle. Triangle BQO an isosceles right triangle, which means it's other two angles are 45 degrees.

    With that information I got this:
    http://img44.imageshack.us/img44/9651/mathhelp.jpg [Broken]

    Knowing the angle, I can find my answer. Thanks a lot to both of you for pushing me in the right direction.


    EDIT: I know the answer now but I have a question about how to write it that one of you may be able to help me with.

    Area of Shaded Part = http://www4b.wolframalpha.com/Calculate/MSP/MSP361966i5a8g78bbbh200003b56bi9282i8195d?MSPStoreType=image/gif&s=10 [Broken] + http://www4b.wolframalpha.com/Calculate/MSP/MSP541966i5a8g757454900001ih9c1ah3gdch55g?MSPStoreType=image/gif&s=10 [Broken] + [PLAIN]http://www4b.wolframalpha.com/Calculate/MSP/MSP1921966hg7d1iggfhie000042hif6h986h19e32?MSPStoreType=image/gif&s=21 [Broken] [Broken]

    I'm studying for an exam in which calculators are not allowed. Is there a way to reduce this further? I can only get to:

    http://www4b.wolframalpha.com/Calculate/MSP/MSP10901966e00a08553fg700004ed479a898634aad?MSPStoreType=image/gif&s=30 [Broken] + [PLAIN]http://www4b.wolframalpha.com/Calculate/MSP/MSP1921966hg7d1iggfhie000042hif6h986h19e32?MSPStoreType=image/gif&s=21 [Broken] [Broken]+ 1
    Last edited by a moderator: May 4, 2017
  8. Jun 21, 2009 #7
    You have there a multiple of a transcendental number added to a multiple of an irrational added to an integer. It's not much of a simplification to complete the addition by finding a common denominator, ie., it's fine the way it is.
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