Welcome to NRICH.

 
Area of an ellipse (oval)
By Mick Chudasama (P2686) on Tuesday, July 4, 2000 - 08:49 pm:

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

By The Editor:

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

By Anonymous on Tuesday, July 4, 2000 - 11:35 pm:

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

x=acos(t)
y=bsin(t)

then integrate ydx from 0 to p/2 and multiply the answer by 4 (because of symmetry). The general answer is Area=pab where a and b are half of the major and minor axes.

Hopefully this diagram shows what is meant by a and b.
By Neil Morrison (P1462) on Wednesday, July 5, 2000 - 12:02 am:

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

By Dan Goodman (Dfmg2) on Wednesday, July 5, 2000 - 01:54 am:

I've always wondered why anyone would integrate, when you can find the result simply by scaling a unit circle in two directions. We know that if a shape S has area A, and we apply the linear transformation T(x,y)=(ax,by), then T(S) has area A.det(T)=abA. Applying this to a circle of unit radius, and an ellipse of semi-minor axis a and semi-major axis b, the area of the ellipse is then ab.(p×12)=pab.

By Neil Morrison (P1462) on Wednesday, July 5, 2000 - 11:45 am:

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

Neil M

By Dan Goodman (Dfmg2) on Wednesday, July 5, 2000 - 03:57 pm:

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.

By The Editor:

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×b=ab square units.

We can do the same thing with a unit circle: one whose radius is 1 unit, and whose area is p×1×1=p 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 pab.