How do you work out the area of an ellipse (oval)?
If you can help I will be very grateful, thank you

There are several explanations below, at very different levels. Don't be put off if you don't understand the first ones!

You just have to integrate the equation of the oval. If the parametric equation of the oval is

$y=b\sin (t)$

then integrate $y\mathrm{dx}$ from 0 to $\pi /2$ and multiply the answer by 4 (because of symmetry). The general answer is Area $=\pi ab$ where $a$ and $b$ are half of the major and minor axes.

Hopefully this diagram shows what is meant by a and b.

The area for a circle must also satisfy the result for an ellipse with a=b, which it clearly does. The canonical equation of an ellipse is:

x² /a² + y² /b² = 1

Yes Dan, but generally students study matrices and transformations after integration. ;)

Neil M

True, but you don't really need to have studied transformations to see that scaling by b in the y-direction will multiply the area by b, similarly in the x-direction.

Putting that into simpler language:

Imagine a square measuring 1 unit by 1 unit (area 1 square unit).
Now imagine we stretch that to be a times as wide. Its area will now be a square units.
Now imagine stretching it to be b times as high. Its area will now be a x b=ab square units.

We can do the same thing with a unit circle: one whose radius is 1 unit, and whose area is pi x 1 x 1=pi square units.
If we stretch it to be a times as wide, and b times as high, its area will be multiplied by a and b, so it will be pi ab.