Final answer:
By counting the lattice points within or on a circle of radius 5 in the first quadrant and multiplying by 4, accounting for symmetry and single-counting axis points, we determine the total number of lattice points within or on the entire circle.
Step-by-step explanation:
To find the number of lattice points within or on a circle with radius 5 centered at (0,0), we can look at the integer coordinate points that satisfy the equation of the circle, which is x2 + y2 = r2. Here, r = 5, so our equation becomes x2 + y2 ≤ 25. We only need to consider the first quadrant and then multiply by 4 (except the origin and points on axes, which don't get multiplied since they don't have counterparts in other quadrants).
Starting with the origin (0,0), which is one point, we consider points (x,y) where x and y are integers. For example, for x = 1, y can be from 0 to 4 (because 12 + 42 = 17 which is less than 25). We find all such pairs, count them, consider symmetry, and aggregate the total number of points.
After a thorough count, we find that there are a total of 21 lattice points in the first quadrant, including (0,0) and points on the x and y axes. Since each quadrant will have the same number of lattice points due to symmetry, and remembering that the points on the axes (except for the origin) should be counted only once, we multiply the number of non-axial points by 4 and add the axial points, including the origin, to get the total number of lattice points.