Undergrad Moon revolution period 27.32, but got 27.53 from equation?

[cnnn]

Hello,

I'm trying to calculate Moon revolution period but always got 27.53 instead of 27.32.
G = 6.674e-11 (m^3 kg^-1 s^-2)
M = 5.9724e24 (kg) from http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html
R = 384400000 (m) Earth Moon distance

From 4π^2 * R^3 = GM*T^2,
got moon revolution period T = sqrt(4π^2 * R^3/(GM))

Earth day in seconds = 23*60*60 + 56*60 + 4
so moon revolution period in Earth day = 27.53

Why it is not 27.32? Is it because the orbit is not circular? But period should be same if R is the semi major axis or the orbit.

If I revise the formula and consider Moon gravitational force to the Earth,
let m be the mass of Moon
T = sqrt(4π^2 * R^3/(G(M+m)))
the moon revolution period in Earth day = 27.36, a bit closer but still some error (rounding error?)

 

[Bystander]

Consider how far the Earth revolves in the lunar month?

 

[cnnn]

Consider how far the Earth revolves in the lunar month?

That would be a synodic month which is 29.53 days.
http://www.sumanasinc.com/webcontent/animations/content/sidereal.html