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Area

The area A ellipse {\displaystyle A_{\text{ellipse}}} enclosed by an ellipse is:

where a {\displaystyle a} and b {\displaystyle b} are the lengths of the semi-major and semi-minor axes, respectively. The area formula π a b {\displaystyle \pi ab} is intuitive: start with a circle of radius b {\displaystyle b} (so its area is π b 2 {\displaystyle \pi b^{2}}) and stretch it by a factor a / b {\displaystyle a/b} to make an ellipse. This scales the area by the same factor: π b 2 (a / b) = π a b. {\displaystyle \pi b^{2}(a/b)=\pi ab.} However, using the same approach for the circumference would be fallacious – compare the integrals ∫ f (x) d x {\textstyle \int f(x)\,dx} and ∫ 1 + f ′ 2 (x) d x {\textstyle \int {\sqrt {1+f'^{2}(x)}}\,dx}. It is also easy to rigorously prove the area formula using integration as follows. Equation (1) can be rewritten as y (x) = b 1 − x 2 / a 2. {\textstyle y(x)=b{\sqrt {1-x^{2}/a^{2}}}.} For x ∈, {\displaystyle x\in,} this curve is the top half of the ellipse. So twice the integral of y (x) {\displaystyle y(x)} over the interval {\displaystyle } will be the area of the ellipse: A ellipse = ∫ − a a 2 b 1 − x 2 a 2 d x = b a ∫ − a a 2 a 2 − x 2 d x. {\displaystyle {\begin{aligned}A_{\text{ellipse}}&=\int _{-a}^{a}2b{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\,dx\\&={\frac {b}{a}}\int _{-a}^{a}2{\sqrt {a^{2}-x^{2}}}\,dx.\end{aligned}}}

The second integral is the area of a circle of radius a, {\displaystyle a,} that is, π a 2. {\displaystyle \pi a^{2}.} So A ellipse = b a π a 2 = π a b. {\displaystyle A_{\text{ellipse}}={\frac {b}{a}}\pi a^{2}=\pi ab.}

An ellipse defined implicitly by A x 2 + B x y + C y 2 = 1 {\displaystyle Ax^{2}+Bxy+Cy^{2}=1} has area 2 π / 4 A C − B 2. {\displaystyle 2\pi /{\sqrt {4AC-B^{2}}}.}

The area can also be expressed in terms of eccentricity and the length of the semi-major axis as a 2 π 1 − e 2 {\displaystyle a^{2}\pi {\sqrt {1-e^{2}}}} (obtained by solving for flattening, then computing the semi-minor axis).

So far we have dealt with erect ellipses, whose major and minor axes are parallel to the x {\displaystyle x} and y {\displaystyle y} axes. However, some applications require tilted ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its emittance. In this case a simple formula still applies, namely

where y int {\displaystyle y_{\text{int}}}, x int {\displaystyle x_{\text{int}}} are intercepts and x max {\displaystyle x_{\text{max}}}, y max {\displaystyle y_{\text{max}}} are maximum values. It follows directly from Apollonios's theorem.