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Math Help - how to create any shape from an area

  1. #1
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    how to create any shape from an area

    Hey all,

    I usually don't deal with heavy maths on a regular basis so the way I communicate this may not be spot on, but here goes.

    I wish to create shapes of ANY number of sides that have a specific area and I know the area of which they have to be.

    For example, the area is 9cm squared, so to create a square shape from that I would squareroot it to find the length and breadth in order to draw the shape. I can also manage to do the same for a triangle by reversing (1/2(bxh)), but the shapes I want to draw are:

    Semicircle - (I think i can do this one)
    Heptagon - 7sides
    dodecagon - 12 sides
    hexadecagon - 16sides
    icosagon - 20 sides

    Any ideas how I create these shapes that have specific areas?


    Thanks very much in advance,
    Dave
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  2. #2
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    There is a formula you can use for the area of a regular polygon.

    Area = \frac{n}{2}\cdot{(radius)^{2}}\cdot{sin(\frac{2\pi  }{n})}. Where n is the number of sides.

    For instance, for the septagon(7 sides), say you want an area of 20 square units:

    \frac{7}{2}r^{2}sin(\frac{2\pi}{7})=20

    Solve the radius and we get r\approx{2.703}

    Then, a side length would be 2rsin(\frac{\pi}{n})=2.703\cdot{2sin(\frac{\pi}{7}  )}=2.346

    I've known these formulas for sometime, but here's a link to a useful site.

    Regular Polygons - Properties
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  3. #3
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    Here is another link that may help.
    Regular Polygon -- from Wolfram MathWorld
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  4. #4
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    excellent

    excellent thanks.

    Not being fluent with advanced maths, how would I type this into a calculator to work it out. (step by step)

    I have no idea what r2 is.

    Thanks.
    Dave
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  5. #5
    Eater of Worlds
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    I mentioned it in my first post. r is the radius, so r^2 is the radius squared.
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  6. #6
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    ahh right, now I see.


    Thanks alot.
    Topic closed I guess.
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  7. #7
    Eater of Worlds
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    Quote Originally Posted by plastiman View Post
    excellent thanks.

    Not being fluent with advanced maths, how would I type this into a calculator to work it out. (step by step)

    I have no idea what r2 is.

    Thanks.
    Dave
    This is not advanced math, just some trig and algebra. Though, I suppose it may seem that way.

    Some calculators have solver functions. Such as a TI-89, TI-92, Voyage 200, etc. To do it algebraically, the old-fashioned way we may proceed as such:

    \frac{7}{2}r^{2}sin(\frac{2\pi}{7})=20

    Multiply both sides by 2/7:

    r^{2}sin(\frac{2\pi}{7})=\frac{40}{7}

    Divide by sin(\frac{2\pi}{7}):

    r^{2}=\frac{40}{7sin(\frac{2\pi}{7})}

    Square root of both sides:

    r=\sqrt{\frac{40}{7sin(\frac{2\pi}{7})}}\approx{2.  703}

    See?.

    Here's a screen capture from the solve function in my TI-92.

    As you can see, it easily solves for r.

    It gives two solutions, but only one is viable because the other is negative.
    Last edited by galactus; November 24th 2008 at 05:38 AM.
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