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From the Fun Fact files, here is a Fun Fact at the Easy level:

Area of an Ellipse

Figure 1
Figure 1

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!

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 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

(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 so-called 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>.

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References:
Subjects:    calculus, analysis
Level:    Easy
Fun Fact suggested by:   Francis Su
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