How would one find the radius of the circle that's the intersection of a sphere and a plane ?
It is some how associated with distance from the center of the sphere.. Distance from plane to center of sphere is 0, then the radius of circle is the radius of the sphere. If distance is equal to radius of sphere then it would simply be a point, radius = 0. Would it be possible to find the radius of the circle using only the distance between the plane and center of sphere?
Also knowing the radius of the sphere.
Best Answer
The answer to your question is yes: Let $O$ denote the center of the sphere (with radius $R$) and $P$ be the closest point on the plane to $O$. Then the distance $OP$ is the distance $d$ between the plane and the center of the sphere.
Now, if $X$ is any point lying on the intersection of the sphere and the plane, the line segment $OP$ is perpendicular to $PX$. So by the Pythagorean theorem we have:
$R^2 = OX^2 = OP^2 + PX^2 = d^2 + PX^2 = d^2 + r^2$
Where $r = PX$ is the radius of the circle of intersection.
So: $r = \sqrt{R^2 - d^2}$