Perimeter of an ellipseGillesCazelaisConsider an ellipse of equationx2a2+y2b2= 1where we assume that 0 < b < a.abyxIt is easy to derive that the area of the ellipse is given by A = ab. Unfortunately there is no simpleformula for the perimeter of an ellipse. We will show that it can be evaluated using an innite series.The ellipse can be represented by the parametric equationsx = a cos and y = b sin, 0 2.Tondtheperimeteroftheellipse,itissucienttondtheperimeterintherstquadrantandmultiply it by four. The perimeter of the ellipse is then given byp = 4_/20_dxd_2+_dyd_2d= 4_/20_a2sin2 + b2cos2 d= 4_/20_a2(1 cos2) + b2cos2 d= 4_/20_a2(a2b2) cos2 d= 4a_/20_1 2cos2 d.Where = _1 b2/a2istheeccentricityoftheellipse. Thisintegraliscalledanellipticintegraland it cant be evaluated using elementary functions.From the binomial formula(1 + x)k= 1 + kx +k(k 1)2!x2+k(k 1)(k 2)3!x3+ , 1 < x < 1we can derive by lettingk = 1/2 that1 + x = 1 +x2 +

n=2(1)n+11 3 5 (2n 3)xn2nn!.1By settingx = 2cos2, we deduce that_1 2cos2 = 1 2cos22

n=21 3 5 (2n 3)2ncos2n2nn!.Note that this series converges for all since 0 2cos2< 1. By integrating termwise this series,we can nd the perimeter of the ellipse.p = 4a_/20_1 2cos22

n=21 3 5 (2n 3)2ncos2n2nn!_d= 4a__/20d 22_/20cos2 d

n=2_1 3 5 (2n 3)2n2nn!_/20cos2n d__.These cosines integrals can be evaluated using Wallis formula_/20cos2n d =12 34 56 2n 12n2=1 3 5 (2n 1)2nn!2, n = 1, 2, 3, . . .By substitution in the above expression forp, we deducep = 4a_2 22_12 2_

n=2_(1 3 5 (2n 3)(2n 1) n)2(2nn!)2(2n 1)2__= 2a_1 _12_22

n=2__1 3 5 (2n 1)2 4 6 2n_22n2n 1__= 2a_1

n=1__1 3 5 (2n 1)2 4 6 2n_22n2n 1__.More explicitly, we obtain the following formula.p = 2a_1 _12_221_1 32 4_243_1 3 52 4 6_265_1 3 5 72 4 6 8_287 _

For example, lets use ve terms in the above series to approximate the perimeter of the ellipsex225 +y216= 1.For this ellipse we havea = 5,b = 4, and =_1 b2/a2= 3/5. We then havep 2a_1 _12_22_1 32 4_243_1 3 52 4 6_265_1 3 5 72 4 6 8_287_ 2(5)_1 _12_2(3/5)2_1 32 4_2(3/5)43_1 3 52 4 6_2(3/5)65_1 3 5 72 4 6 8_2(3/5)87_ 28.36.The ellipse has a perimeter of about 28.36 units.2