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 \[2 \times 0 = 0\]