Given the polar equation of a limacon $r=2+2\cos \theta$, sketch the graph of the limacon and find the length of its inner loop.

Step 1 ：Step 1: Identify the type of limacon. Here, since the coefficient of $\cos \theta$ is equal to the constant, it is a limacon with an inner loop.

Step 2 ：Step 2: Sketch the graph. We can plot a few values of $r$ for various values of $\theta$ and then smoothly connect the points to form the limacon. The graph will show a loop within the main body of the limacon.

Step 3 ：Step 3: Find the length of the inner loop. This is twice the value of $r$ when $\theta =\pi$, because at this angle, the distance from the pole to the limacon is at its minimum. Substitute $\pi$ into the equation: $$r=2+2\cos \pi =2-2=0$$ So the length of the inner loop is $$2\times 0=0$$