Squares on Spheres ```Name: James K. Status: student Age: N/A Location: N/A Country: N/A Date: N/A ``` Question: A square on the surface of a sphere measuring 1 degree 1/360th. of a circle ) square. Replies: How many of these squares cover the surface of that sphere? There is a fundamental problem here. If you try to paste a postage stamp on a soccer ball, it won't lay flat. That's why soccer balls have a pentagonal surface. But assuming that the squares are small enough that the above problem is negligible, here's how the problem goes: The surface of a sphere is S=4*pi*r^2, where pi=3.14... and we take r=1 So the total surface area S=4*pi(radians^2). Now 360 (deg) = 2*pi(radians), so 360^2 (deg^2) = (2*pi)^2 (radians) = 4*pi^2(radians^2). So the total surface area S=4*pi (radians^2) * (360)^2 (degrees^2)/ 4*pi^2 (radians^2). Canceling out 4*pi in the numerator and denominator leaves: S = (360)^2/ pi (degree^2) = 41253 (degree^2) patches 1(degree)x1(degree). This number is rounded to the nearest (degree^2) because no integer number of tiny squares can evenly and completely cover a sphere. Vince Calder James, I cannot tell exactly how many will fit due to the fact that there might be a little bit of open space between some of the circles. A way to get a very good approximation, and definitely a maximum, involves area. Choose a radius. The side of a square will be 1/360th of the distance around a circle. Calculate the area of such a square. Calculate the surface area of a corresponding sphere: 4(pi)(radius)^2. See how many square areas will fit into a sphere area. Dr. Ken Mellendorf Click here to return to the Mathematics Archives

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