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Math Help - A Circle Circumscribed about a hexagon

  1. #1
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    A Circle Circumscribed about a hexagon

    i need your help again !
    help me to solve this...please
    God bless you more .


    *A circle is circumscribed about a hexagon. The area outside the hexagon but inside the circle is 15 square meters.

    a) compute the radius of the circle.
    b)compute the area of the hexagon.
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  2. #2
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    Quote Originally Posted by arenkun View Post
    i need your help again !
    help me to solve this...please
    God bless you more .


    *A circle is circumscribed about a hexagon. The area outside the hexagon but inside the circle is 15 square meters.

    a) compute the radius of the circle.
    b)compute the area of the hexagon.
    HI

    I assume its a regular hexagon .

    So the angle each angle at the centre would be 60 degree .

    The formula for area of a segment , A=\frac{1}{2}r^2(\theta-\sin \theta)

    Since there are 6 segments , 15=6(\frac{1}{2}\times r^2(\frac{\pi}{3}-\frac{\sqrt{3}}{2}))
    evaluate r which is approximately 5 m .

    Then the area of the hexagon would be the sum of all the 6 triangles ,

    Area =6(\frac{1}{2}\times 5\times 5\times \sin 60)
    Last edited by mathaddict; January 12th 2010 at 11:00 PM.
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  3. #3
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    Hello, arenkun!

    A circle is circumscribed about a hexagon.

    The area outside the hexagon but inside the circle is 15 mē.

    (a) Compute the radius of the circle.

    (b) Compute the area of the hexagon.
    Code:
                    A
                  * o *
              *           *
            *               *
         F o                 o B
    
          *                   *
          *         *         *
          *         O         *
    
         E o                 o C
            *               *
              *           *
                   * o *
                     D

    The center of the circle is O.
    The vertices of the hexagon are: . A,B,C,D,E,F.
    Draw chords: . AB, BC, CD, DE, EF, FA
    . . .and radii: . OA, OB, OC, OD, OE, OF

    Assuming it is a regular hexagon, all the chords and radii are equal to the radius r.


    The area of the circle is: . \pi r^2


    The hexagon is comprised of six congruent equilateral triangles of side r.
    The area of one triangle is: . \frac{\sqrt{3}}{4}r^2
    The area of the hexagon is: . 6 \times\frac{\sqrt{3}}{4}r^2 \:=\:\frac{3\sqrt{3}}{2}\,r^2


    The difference of the areas is 15 mē: . \pi r^2 - \frac{3\sqrt{3}}{2}r^2 \:=\:15

    . . \left(\pi - \frac{3\sqrt{3}}{2}\right)r^2 \:=\:15 \quad\Rightarrow\quad \left(\frac{2\pi - 3\sqrt{3}}{2}\right)r^2 \:=\:15 <br />

    Hence: . r^2 \:=\:\frac{30}{2\pi-3\sqrt{3}} \quad\Rightarrow\quad r \;=\;\sqrt{\frac{30}{2\pi-3\sqrt{3}}}


    Therefore: . r \;=\;5.253385683\text{ m} \;\;(a)



    The area of the hexagon is: . \frac{3\sqrt{3}}{2}\,r^2 \;=\;\frac{3\sqrt{3}}{2}\left(\frac{30}{2\pi-3\sqrt{3}}\right) \;=\;71.70186611\text{ m}^2\;\;(b)

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  4. #4
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    TO : Math Addict and Soroban


    I really thank you guys for helping me.
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