 Therefore, a relatively reasonable approximation for the length and period are,

where is the number of seconds between "two" beats (one beat for each side of the swing), and is measured in metres.For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method (),

and then integrating over one complete cycle,

or twice the half-cycle

or four times the quarter-cycle

which leads to

Note that this integral diverges as approaches the vertical

so that a pendulum with just the right energy to go vertical will never actually get there.