The number of points with integral coordinates that are interior to the circle

$x^2+y^2=16$  is

The number of points is equal to the number of integral solutions $\left(x,y\right)$  such that $x^2+y^2<16.$ So $x,y$  are integers such that $-3\le x\le 3,-3\le y\le 3$  satisfying the inequation $x^2+y^2<16.$

The number of selections of values is 7, namely –3, –2, –1, 0, 1, 2, 3.

The same is true for y. So the number of ordered pairs $\left(x,y\right)$  is $7\times 7$ .

But $\left(3,3\right),\left(3,-3\right),\left(-3,3\right),\left(-3,-3\right)$ are rejected because they do not satisfy the inequation $x^2+y^2<16.$

So, the number of points is 45.