Let *a* be the distance from the point (0,0) to a point on the figure (known as
a cardoid). Next consider the angle t formed from the points (-1,0), (0,0) and *a* at (0,0).

After working through the trigonomtry (this step is left to you) the length of a in terms of t is 2*(1-cos(t)).

The formula for arc length is the integral of
( a^{2} + (da/dt)^{2} )^{1/2}.

In this case it is the integral from 0 to 2*pi of:

( 4*(1-cos(t))^{2} + 4*sin^{2}(t) )^{1/2}. =

8^{1/2} * (1 - cos(t))^{1/2}.

After consulting a table of integrals we find that the integral of
(1 - cos(t))^{1/2} is -2*2^{1/2}*cos(t/2).

Taking the integral the answer works out to 16.

I'd like to thank Andrea for correcting an earlier mistake in my answer.

Michael Shackleford, A.S.A.