How many coordinates are necessary to determine a sphere? Do determine a circle, you would need at least three coordinates. How many are necessary to determine a sphere?
 A: I believe the answer is 4 (and hopefully someone will correct me if I'm wrong...).  Here's why.  Any 3 points are coplanar.  Consider 3 points on the plane A, B, and C.  One can connect each pair of points with a line segment to create a triangle, then circumscribe a circle.  The center of said circle is equidistant to all 3 points.  Consider a line passing through the center perpendicular to this plane.  It is easy to prove that any point on this line is also equidistant from A, B, and C.  Therefore, it must take more than 3 points to determine a sphere.
A: 1st Approach:
The first thought you would have is that 3 points are sufficient to describe a circle and after rotating the circle about its diameter, you would get a sphere. But this is the special case when the circle you choose is itself an equator of the sphere and the center of the 'Circle' is also the center of the 'Sphere'.
 However if the circle is different than the equator, in that case you need to know the location of the center of sphere to fully define it, along with the three points to define the circle.
So,
4 points(sphere) = 3 points(circle) + 1 point(center of sphere)
2nd Approach:
Equation of sphere:
$$(x-a)^2 + (x-b)^2 + (x-c)^2 = R^2  $$
You have 4 variables: $a,b,c,R$.
So you need 4 equations(4 points) to determine a sphere.
A: A nice explanation for 4. 3 gives you a circle. But you need one more to determine the size of the circle compared to the sphere.
