Calculate area of Ellipse without calculus?

Question

I like the way integration works, but the final formula $\pi ab$ is too simple. I know there is a more deeper way to derive it. I just don't like to use calculus here, too many equations.

I'd like to use simple math, which does offer deeper insight into it.

Answer 1

Think about it this way. You start off with a circle of radius $a$ of which you know that it has area $\pi \cdot a^2$. Now you pick a direction (say horizontally for concreteness) and stretch the circle in that direction so that what used to be the diameter of length $2a$ will afterwards have length $2b$. Consequently, every line that lies horizontally will have been stretched by a factor of $b/a$, while you leave the vertical direction invariant. Then your total area will also have been changed by a factor of $b/a$, hence yielding $\pi \cdot a \cdot b$.

Answer 2

Consider the unit disk (bounded by the circle of radius $1$, centered at the origin). Now, to construct an ellipse whose axes are $a$ along the $x$-axis and $b$-along the $y$-axis. This corresponds to the application of the linear transformation $$ \begin{bmatrix}a&0\\0&b\end{bmatrix}. $$ We can confirm that this is an ellipse because if your original coordinates are $x_1$ and $x_2$ while your new coordinates are $y_1$ and $y_2$, we have $y_1=ax_1$ and $y_2=bx_2$. Therefore, $y_1$ and $y_2$ satisfy: $$ \frac{y_1^2}{a^2}+\frac{y_2^2}{b^2}=1. $$

Since linear transformations scale areas by the determinant (and the original disk has area $\pi$), the resulting area is $ab\pi$.

Answer 3

You may use an affine map $\varphi$ to send an ellipse into a circle. Since affine maps preserve the ratios between areas, the area of the ellipse is $\frac{\text{Area}(\text{circle})}{\left|\det\varphi\right|}=\pi a b$.

Answer 4

Draw a rectangle of known area around your shape. Start throwing a known number of darts at the rectangle randomly, the more the better. Count the number of darts landed inside the boundaries of your shape versus those outside, but still within the boundaries of the rectangle. Apply that ratio to the known area of the rectangle, and there you have an approximation of the size of your shape.