Lattice Points Problem Part 2: Circles, Gauss' Circle Problem and Pick's Theorem

Eric Jablow inspired me to develop the following extended enrichment actvity/project for Geometry students. You can research the general solution of Gauss' Circle Problem in MathWorld but for this application it isn't necessary to use that formula.

Consider the circle of radius 5 centered at the origin.

(a) Determine the coordinates of the 12 lattice points on this circle. Recall that lattice points are points whose coordinates are both integers.
Note: Is it reasonable from symmetry arguments that the number of lattice points is divisible by 4?
(b) Using graph paper, show there are 69 lattice points INSIDE this circle. Describe your counting method.

The total number of lattice points inside or on our circle is 69+12 = 81. This agrees with the result from Gauss' formula but we will now 'approximate' this result using Pick's Theorem which gives the relationship among interior, boundary points and the area of a polygon whose vertices are lattice points.

(c) Consider the inscribed dodecagon formed by connecting the 12 lattice points from part (a). Determine the lengths of the sides of this polygon.

(d) By dividing the polygon into 12 triangles show that the area of this polygon is 74. No trigonometry, just Pythagorean and basics!
Hint: This is not a regular polygon but you can still divide it into 12 isosceles triangles.
Comment: Do you find it surprising that the area is rational (in fact, integral), considering that the sides are irrational?

(e) Pick's Theorem states that the area of a polygon whose vertices are lattice points is given by the formula A = I + B/2 - 1 where I = the number of interior lattice points and B = the number of boundary lattice points, that is, points on the polygon.
Show that Pick's Theorem leads to I = 69.

(f) For our problem the number of lattice points inside the circle matched the number of points inside the inscribed polygon. A coincidence? Whether it's true or not, explain why this result seems to make sense.

(g) To further investigate this 'coincidence', change the radius to 10.
(i) Show, by counting, that there are 317 lattice points inside or on this circle.
(ii) Show that there are still 12 lattice points on this larger circle.
(iii) Show that there are 303 lattice points inside or on the resulting dodecagon.
[As before, find the area w/o trig and use Pick's Thm. Note: To find the area of the polygon use (d) and ratios!]
(iv) Show that the point (4,9) is inside this circle but outside the dodecagon! This suggests why Pick's theorem fails in this case! Why?

Good luck! This is an extended challenge that I will leave up for several days and invite comment. Whether you implement it in a classroom or not, enjoy!

[By the way, some of the numerical results (like 303) have not been independently verified. If you find an error, let me know!]