Intersection of a Plane and a Sphere

Intersection of a Plane and a Sphere

How to solve the intersection of a plane and a sphere problem: property, example, and its solution.

Property

The intersection of a plane and a sphere is a circle.

The intersection of a plane and a sphere is a circle.

The segment that connects
the center of the sphere
and the center of the circle
is perpendicular to the intersecting circle.

This is the 3D version of the chord of a circle.

Example

Plane alpha intersects the sphere in the blue circle. The distance between the center of the circle of the sphere and the plane is 3. Find the area of the circle. The radius of the sphere: 5.

The distance between the sphere and the circle is 3.

So the segment that connects
the center of the sphere
and the center of the blue circle is 3.

Next, the radius of the sphere is 5.
So draw a radius
whose endpoint is on the endpoint of the blue circle.

See the right triangle in the sphere.
Its hypotenuse is 5.
And its leg is 3.

So it's a (3, 4, 5) triangle.

So the radius of the blue circle is 4.

So the area of the blue circle is π⋅42.

Area of a circle