A circle of radius 5 with its center at $(0,0)$ is drawn on a Cartesian coordinate system. How many lattice points (points with integer coordinates) lie within or on this circle?
The table below shows for each value of $x$ which values of $y$ satisfy the condition that $(x,y)$ lies on or inside the circle of radius 5 centered at the origin. \begin{tabular}{ccc}
$x$ & Constraints & Number of $y$ values \\
$\pm5$ & $y=0$ & 1 \\
$\pm4$ & $-3\leq y \leq 3$ & 7 \\
$\pm3$ & $-4\leq y \leq 4$ & 9 \\
$\pm2$ & $-4\leq y\leq 4$ & 9 \\
$\pm1$ & $-4\leq y\leq 4$ & 9 \\
0 & $-5\leq y\leq 5$ & 11 \\
\end{tabular} In total, there are $2(1+7+9+9+9)+11=81$ lattice points on or inside the circle.