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Question:
Is there an equation that can determine how many squares or rectangles will fit into a circle. e.g.. How many 25" x 30" rectangles will fit into a 120" diameter circle



Replies:
Either 1 or 0. Consider the following geometric construction: A circle of radius = 1, centered at the origin of the (X,Y) plane, i.e. at (0,0). Draw any diameter of the circle through the origin (0,0) at an angle, 'a', with respect to the positive 'X' axis in the usual counterclockwise direction. Where this diameter intersects the circle draw a line parallel to the 'Y' axis. This line will intersect the circle in the lower right quadrant. Draw a line from that point from right to left parallel to the 'X' axis. This construction makes a right triangle with a vertical side = 2*sin(a) and a horizontal side = 2*cos(a). The diameter forms the hypotenuse of the triangle and has length 2. So, by the Pythagorean theorem: 2*2 = (2*cos(a))*(2*cos(a)) + (2*sin(a))*(2*sin(a)) or 1 = [cos(a)]^2 + [sin(a)]^2.

The bottom line is (bad pun). Once you have chosen the angle 'a' and the radius 'R' the rest of the geometry is determined.

Vince Calder



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