(Mail to some kids.)
Beginners may find it discouraging that the circumference of an ellipse requires this unfamiliar EllipticE function, but it is actually well worth familiarizing! For example, it provides the world's most rapidly convergent π formulas, and has fascinating properties.
Actually, there are infinitely many eccentricities where the circumference EllipticE is expressible in terms radicals and factorials, but they're factorials of fractions! You may already know that In[210]:= (1/2)! Out[210]= √π/2 but halves are the only known fractions whose factorials are familiar. Somewhat amazingly, the 1 × 1/√2 ellipse (bounding box 2×√2) has circumference 9 π^(3/2)/(16 (3/4)!^2) + 32 (3/4)!^2/(9√π) ~ 5.4025755241907 (compared with the perimeter of the bounding box = 6.82842712474619.) But this is perhaps the nicest case. The circumference of a 1 by (1/𝝓 + 1/√𝝓)/√2 ellipse is (where 𝝓 := (1+√5)/2, the Golden Ratio) ArcLength[Circle[{,}, {(1/𝝓 + 1/√𝝓)/√2, 1}]] == 9 π^(3/2)/(10 √2 5^(7/8) 𝝓^(1/4) (1/20)! (9/20)!) + 2 √2 5^(3/8) (4 √5 + 10 √𝝓) (1/20)! (9/20)!/(9 𝝓^(1/4) √π) ~ 6.26092807313208, 2π-ish because In[255]:= N[(1/GoldenRatio + 1/√GoldenRatio)/√2] Out[255]= 0.992908994700242 That's rounder than an Indiana circle. —rwg