How Many Integer Lattice Points Are Inside a Circle?

A lattice point is a point whose coordinates are both integers. For a circle centered at the origin, counting the lattice points inside it means counting integer pairs $(x,y)$ whose distance from the origin is no greater than the radius.

The area of the circle, $\pi R^2$, gives a useful estimate, but it is not generally the exact count. The exact count can be found by checking the possible integer $x$-coordinates and counting the allowed $y$-coordinates in each vertical column.

Exact formula for a centered circle

Let the circle have radius $R\geq 0$ and equation $x^2+y^2=R^2$. For a fixed integer $x$, the points in the disk satisfy

$$y^2\leq R^2-x^2.$$

If $|x|\leq R$, the permitted integer values of $y$ run from $-\lfloor\sqrt{R^2-x^2}\rfloor$ through $\lfloor\sqrt{R^2-x^2}\rfloor$. Thus that column contributes $2\lfloor\sqrt{R^2-x^2}\rfloor+1$ points. Summing over all possible integer $x$ gives

$$N(R)=\sum_{x=-\lfloor R\rfloor}^{\lfloor R\rfloor}\left(2\left\lfloor\sqrt{R^2-x^2}\right\rfloor+1\right).$$

This formula counts points inside or on the circle. It applies even when $R$ is not an integer.

Worked example: radius 5

For $R=5$, the integer $x$-coordinates range from $-5$ to $5$. Because the disk is symmetric about the vertical axis, count $x=0$ once and double the counts for positive $x$:

$x$-coordinate	Allowed integer $y$-coordinates	Points in column
$0$	$-5$ through $5$	$11$
$1$	$-4$ through $4$	$9$
$2$	$-4$ through $4$	$9$
$3$	$-4$ through $4$	$9$
$4$	$-3$ through $3$	$7$
$5$	$0$	$1$

Therefore,

$$N(5)=11+2(9+9+9+7+1)=81.$$

So there are 81 integer lattice points inside or on the circle of radius 5. If “within” means strictly inside, exclude the 12 points on the circumference: $(\pm5,0)$, $(0,\pm5)$, and the eight points obtained by changing the signs of $(3,4)$. The strict count is $81-12=69$.

Connection to sums of two squares

Another way to organize the count is by distance from the origin. A lattice point on the circle of radius $\sqrt n$ corresponds to an integer solution of $x^2+y^2=n$. Let $r_2(n)$ be the number of such ordered integer pairs, with signs and coordinate order counted. Then the number inside or on a radius-$R$ circle is

$$N(R)=1+\sum_{n=1}^{\lfloor R^2\rfloor}r_2(n),$$

where the extra $1$ counts the origin. This viewpoint separates the points on each individual circle (each value of $n$) from the points in the whole disk.

For $n\geq1$, a classical formula is $r_2(n)=4(d_1(n)-d_3(n))$, where $d_1(n)$ and $d_3(n)$ count the positive divisors of $n$ congruent to 1 and 3 modulo 4, respectively. For example, the positive divisors of 25 are 1, 5, and 25, all congruent to 1 modulo 4. Thus $r_2(25)=4(3-0)=12$, matching the 12 lattice points on the radius-5 circumference.

If the circle has a different center

For a circle centered at $(a,b)$ with radius $R$, the condition is $(x-a)^2+(y-b)^2\leq R^2$. For each integer $x$ with $|x-a|\leq R$, the allowed integer $y$ values run from

$$\left\lceil b-\sqrt{R^2-(x-a)^2}\right\rceil \quad\text{to}\quad \left\lfloor b+\sqrt{R^2-(x-a)^2}\right\rfloor.$$

Count the integers in that range for each such $x$ and add them. The centered-circle formula is a special case with $a=b=0$.

Common mistakes

• Using area as the exact count: $\pi R^2$ estimates the number of lattice points, but it need not be an integer and usually is not the exact answer.
• Forgetting the floor: The largest permitted integer $y$ is the floor of the square root, not the square root itself.
• Mixing up boundary conventions: “Inside” may mean strictly inside, while “inside or on” includes the circumference. State which count you want.
• Counting only positive coordinates: The formula counts negative coordinates and zero as well. For example, $(3,4)$ and $(-3,4)$ are distinct lattice points.