Figure 1

You know the formula for the area of a circle of radius R.
It is Pi*R^{2}.
But what about the formula for the area of an ellipse of
semimajor axis of length A and
semiminor axis of length B? (These semimajor axes are half the lengths of, respectively, the largest and smallest diameters of the ellipse see Figure 1.)
For example, the following is a standard equation for such an ellipse centered at the origin:
(x^{2}/A^{2}) + (y^{2}/B^{2}) = 1.
The area of such an ellipse is
Area = Pi * A * B ,
a very natural generalization
of the formula for a circle!
Presentation Suggestions:
If students guess this fact, ask them what they think
the volume of an ellipsoid is!
The Math Behind the Fact:
One way to see why the formula is true is to realize
that the above ellipse is just a unit circle that has been
stretched by a factor A in the xdirection, and a factor
B in the ydirection. Hence the area of the ellipse
is just A*B times the area of the unit circle.
The formula can also be proved
using a trigonometric substitution.
For a more interesting proof, use line integrals
and Green's Theorem in multivariable calculus.
Each of the above proofs will generalize to
show that the volume of an ellipsoid with
semiaxes A, B, and C is just
(4/3)*Pi*A*B*C.
(Just think of a stretched sphere, use trig substitution, or use
an appropriate flux integral.)
By the way, unlike areas, the formula for the length of the perimeter of a circle does not generalize in any nice way to the perimeter of an ellipse, whose arclength is not expressible in closed form this difficulty gave rise to the study of the socalled elliptic integrals.
How to Cite this Page:
Su, Francis E., et al. "Area of an Ellipse."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
