jmc

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{array}{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{array} In total, there are $2(1+7+9+9+9)+11=\boxed{81}$ lattice points on or inside the circle.