Dicey Diameters

To have the area of the circle be less than the circumference, we must have $\pi (\frac{d}{2})^{2}< \pi d$ where $d$ is the circle's diameter.

Solving for $d$ , we get $d< 4$ .

Since $d$ is the sum of two dice, the only possible values of $d$ are 2 and 3.

To roll a sum of 2: you must roll two 1’s. $\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}$

To roll a sum of 3: you must roll one 1 and one 2. There are two different ways to do this. $2\times \frac{1}{6}\times \frac{1}{6}=\frac{2}{36}$

Add these up to get the final answer. $\frac{1}{36}+\frac{2}{36}=\frac{3}{36}=\frac{1}{12}$

$1+12 = 13$ .

The answer is $\boxed{13}$ .