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Math Help - find area of shaded area (yellow)

  1. #1
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    find area of shaded area (yellow)

    2 quadrants within a square of length 10cm.

    how to find the area of the shaded area in yellow?

    thanks!

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  2. #2
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    Krizalid's Avatar
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    Find an equilateral triangle. There's a lot of hidden information there.
    Last edited by Krizalid; September 2nd 2007 at 10:12 AM.
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  3. #3
    MHF Contributor red_dog's Avatar
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    You can split the area in two sectors with the same area and an equilateral triangle.
    Each sector has an angle of 30^{\circ}.
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  4. #4
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    Hello, bells!

    Did you catch red_dog's excellent hint?


    Two quadrants within a square of length 10cm.
    . . Find the area of the shaded region.



    Label the square A\!-\!B\!-\!C\!-\!D, starting at the upper-left and labelling clockwise.
    Label the intersection of the two arcs with O.
    Draw radii OC and OD.

    Triangle OCD is equilateral with side 10cm.
    . . I'm sure you can find its area.

    Sectors DAO and CBO have radius 10 and central angles 30.
    . . and you can find their areas, too, right?

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  5. #5
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    why is it an equilateral triangle?

    thanks.
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  6. #6
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    Quote Originally Posted by bells View Post
    why is it an equilateral triangle?

    thanks.
    Hello,

    because all three sides have the same length.

    I used
    - your drawing
    - red_dog's suggestion
    - Sorobans labeling

    see attachment
    Attached Thumbnails Attached Thumbnails find area of shaded area (yellow)-gelb_flaeche.gif  
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by bells View Post
    why is it an equilateral triangle?

    thanks.
    earboth gave you detailed information on how to find the answer, but allow me to fill in a bit of the background.

    recall that the yellow regions are outlined by circular arcs with radius 10. consider the left arc. any line drawn from the bottom left corner to any point along the arc will have the length of the radius, which is 10. the same thing holds true for the arc on the right. since the base of the triangle drawn is one side of the square, it will have a length of 10 as well. thus we have a triangle with sides 10 units of length each, thus it is an equilateral triangle
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  8. #8
    Bar0n janvdl's Avatar
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    Wow, this was an excellent problem. I like it.
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  9. #9
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    got it!

    thanks!
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