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Math Help - ellipse area

  1. #1
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    ellipse area

    i know that an ellipse area can be easily computed by integration
    however i wonder if there might be ageometric way to get the same result?
    i tried really hard but all i got were bad approximations
    any ideas? thank in advance!
    best regards
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: ellipse area

    Without integration you can start from the area of a circle wich is given by: A=\pi \cdot r^2
    The cartesian equation of an ellips is:
    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1
    and if a=b then we get a circle with radius a. That means the area of an ellips is:
    A=\pi\cdot a\cdot b, because if a=b (if you've a circle) you get the area of a circle with radius a: A=\pi\cdot a^2

    Is this what you're looking for? ...
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  3. #3
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    Re: ellipse area

    thanks alot
    its almost what i am looking for
    actually i thought there might be ageometric process
    involving limits
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: ellipse area

    I don't know, limits are used voor approximations. Can you be more specific? Are you searching for a formula of the area of an ellips which is using limits?
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  5. #5
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    Re: ellipse area

    Quote Originally Posted by islam View Post
    thanks alot
    its almost what i am looking for
    actually i thought there might be ageometric process
    involving limits
    Not sure if you are looking for this one:

    1. Consider a quarter circle. The area of the quarter circle can be approximated by the sum of areas of small rectangles whose width is \Delta x and whose length is the appropriate y-value ( = y_C). Divide the length a into n equal parts (that means \Delta x = \frac an ) then you get the area of the quarter circle by:

    A_{\tfrac14c} = \lim_{\Delta x \to 0}\left(\sum_{i=1}^n y_C_i \cdot \Delta x\right) = \frac14 \cdot \pi \cdot a^2

    2. An ellipse is produced by a perpendicular dilation (not sure if this is the correct expression) with a fixed quotient:

    \dfrac{y_E}{y_C} = \dfrac ba~\implies~y_E=\dfrac ba \cdot y_C

    3. The area of the quarter ellipse is consequently calculated by:

    A_{\tfrac14e} = \lim_{\Delta x \to 0}\left(\sum_{i=1}^n y_E_i \cdot \Delta x\right) = \lim_{\Delta x \to 0}\left(\sum_{i=1}^n \frac ba \cdot y_C_i \cdot \Delta x\right)=\frac ba \cdot \lim_{\Delta x \to 0}\left(\sum_{i=1}^n y_C_i \cdot \Delta x\right)= \frac ba \cdot \frac14 \cdot \pi \cdot a^2=\frac14 \cdot \pi \cdot b \cdot a

    4. Determine the area of a complete ellipse.
    Attached Thumbnails Attached Thumbnails ellipse area-ellipsflauskrsflche.png  
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  6. #6
    Grand Panjandrum
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    Re: ellipse area

    Quote Originally Posted by Siron View Post
    Without integration you can start from the area of a circle wich is given by: A=\pi \cdot r^2
    The cartesian equation of an ellips is:
    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1
    and if a=b then we get a circle with radius a. That means the area of an ellips is:
    A=\pi\cdot a\cdot b, because if a=b (if you've a circle) you get the area of a circle with radius a: A=\pi\cdot a^2

    Is this what you're looking for? ...
    This is a non sequitur in that the conclusion does not follow from the premise, at least with the argument presented. You need the observation that the ellipse is a dilation of the exterior tangent circle along the minor axis, see earboth's post for details.

    CB
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