Area of an Oval

Date: 11/30/2001 at 17:32:40
From: Colin Franks
Subject: Area of an Oval

How do I figure out the AREA of an oval? The oval in question is
17" x 38" and I need to know the total square feet. Or, more 
specifically, how much bigger (proportionally) I can make this oval to 
achieve 4.3 sq. ft.

Thank you.
Colin

Date: 11/30/2001 at 19:15:06
From: Doctor Paul
Subject: Re: Area of an Oval

In mathematics, an "oval" is more correctly called an ellipse, and the 
area of an ellipse is given by Pi*a*b, where 2*a is the length of the 
longer side (measured at its greatest width) and 2*b is the length of 
the shortest side (measured at its greatest height).  

Here's a picture of how to measure, from Eric Weisstein's World of 
Mathematics::

   Ellipse
   http://mathworld.wolfram.com/Ellipse.html   

In your case, we have a = 19 inches and b = 8.5 inches.

Since you want the area to be in square feet, let's do the conversion:

   a = 19/12 feet and b = 8.5/12 feet

Thus the area of this ellipse is Pi*19*8.5/144 = 3.52 square feet.

That's not what you need, though, since we need to somehow expand a 
and b while keeping the relation:

   a/b = 38/17

This says that a = 38*b/17

Using this formula, we want the area to be 4.3 sq feet, so

   Pi*a*b = 4.3

Substituting for a gives:

   Pi*(38*b/17)*b = 4.3

so that says:

   b^2 = 4.3*17/(38*Pi) = .61233 ft

   b = .7825 ft = 9.39 inches

plugging in b = 9.39 in the formula a = 38*b/17 gives 

   a = 20.9898 inches

Thus the dimensions of your oval should be 

   9.39*2 x 20.9898*2

or

   18.78 inches x 41.98 inches

I hope this helps.  Please write back if you'd like to talk about this 
some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/