Area of circle formed when sphere is sliced by a plane

Question

First off, when a sphere is cut by a plane, is a circle always formed or does a ellipse get formed in some cases? If a circle is always formed, how do you prove it?

Next, how would you find the area of the circle formed when the sphere described by the equation $$x^{2}+y^{2}+z^{2}\; =\; 5$$

is cut by the plane $$x+y+z=1$$? I know how you would do it if it was just a vertical or horizontal plane, but I am unsure of how to proceed in this case. I think this system of two equation with three variables must be simplified into the equation of a circle but am unsure of how to do it.

Answer 1

Try putting $y=0$ and drawing the cross-section of the problem in the $xz$-plane. You can find that the line $x + z = 1$ intersects the circle $x^2+z^2=5$ in two points, $(-1, 2)$ and $(2,-1)$. The line joining these two points is a diameter of the desired circle. Compute the length of the diameter, and you can find the area of the circle.

Answer 2

Find the distance of the center of the sphere to the plane $D$ using the formula $D=|\frac{ax+by+cz+d}{\sqrt{a^2+b^2+c^2}}|$ where $(x,y,z)$ is the center, $a,b,c,d$ are the constants in the equation of the plane $ax+by+cz+d=0$.
The radius of circle is $\sqrt{r_s^2-D^2}$.
This is because the line perpendicular to the plane and a radius of the sphere to the circle forms a right angled triangle. Hence applying Pythagorus Theorem ( $h^2=s_1^2+s_2^2$) with radius of circle as the hypotenuse.