You know the formula for the area of a circle of radius R.
It is Pi*R2.
But what about the formula for the area of an ellipse of
semi-major axis of length A and
semi-minor axis of length B? (These semi-major 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:
(x2/A2) + (y2/B2) = 1.
The area of such an ellipse is
Area = Pi * A * B ,
a very natural generalization
of the formula for a circle!
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 x-direction, and a factor
B in the y-direction. 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
semi-axes A, B, and C is just
(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 so-called elliptic integrals.
How to Cite this Page:
Su, Francis E., et al. "Area of an Ellipse."
Math Fun Facts.