MHB Calculating the Number of Squares Inside a Circle in the 1st Quadrant

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A 10 by 10 square contains 100 1 by 1 squares, and a circle drawn inside is tangent to all sides. The discussion focuses on calculating how many of these squares are fully inside the circle, with an initial estimate of 60 squares. A proposed method involves programming a loop to count the squares in the first quadrant based on the circle's radius. The formula iterates through x values, calculates corresponding y values, and multiplies the count by four to account for all quadrants. The accuracy of this method and the need for a general case formula are key points of consideration.
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A 10by10 square contains 100 1by1 squares (of course!).
A circle is drawn inside above square, tangent to all 4 sides.
How many of the 1by1 squares are fully inside the circle?

I get 60...which I think is correct.
Trying to devise a general case formula...
 
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Couldn't come up with general case formula.

However, this simple program seems to work:

-center of circle at origin
-examine the 1/4 circle in 1st quadrant

r = radius
x,y = points on circumference

INPUT r
LOOP x FROM 1 TO r
y = FLOOR[SQRT(r^2 - x^2)]
count = count + y
ENDLOOP
PRINT count*4

See anything wrong?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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