1. ## 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
best regards

2. ## 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? ...

3. ## Re: ellipse area

thanks alot
its almost what i am looking for
actually i thought there might be ageometric process
involving limits

4. ## 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?

5. ## Re: ellipse area

Originally Posted by islam
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.

6. ## Re: ellipse area

Originally Posted by Siron
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