Equation of a Sphere (Cartesian Form)

In three-dimensional geometry, a sphere is a set of points that are all at the same distance from a fixed point in space. This fixed point is called the center of the sphere, and the constant distance is the radius of the sphere. The equation of a sphere in Cartesian coordinates describes all the points (x, y, z) that lie on the surface of the sphere.

Standard Equation of a Sphere

The standard equation of a sphere with center at the origin (0, 0, 0) and radius r is given by:

[ x^2 + y^2 + z^2 = r^2 ]

This equation is derived from the Pythagorean theorem applied in three dimensions.

General Equation of a Sphere

If the center of the sphere is at a point (h, k, l) and the radius is r, the equation of the sphere is:

[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 ]

This equation represents a sphere that is translated from the origin to the point (h, k, l).

Table of Differences and Important Points

Feature	Sphere at Origin	Sphere at (h, k, l)
Center	(0, 0, 0)	(h, k, l)
Equation	x^2 + y^2 + z^2 = r^2	(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
Radius	r	r
Symmetry	Symmetric about the origin	Symmetric about the point (h, k, l)

Formulas

• Volume of a Sphere: The volume V of a sphere with radius r is given by:

[ V = \frac{4}{3} \pi r^3 ]

• Surface Area of a Sphere: The surface area A of a sphere with radius r is:

[ A = 4 \pi r^2 ]

Examples

Example 1: Sphere at the Origin

Find the radius of a sphere with the equation x^2 + y^2 + z^2 = 36.

Solution: The given equation is in the standard form with r^2 = 36. Therefore, the radius r is:

[ r = \sqrt{36} = 6 ]

Example 2: Sphere with Center at (h, k, l)

Determine the center and radius of the sphere with the equation (x - 2)^2 + (y + 3)^2 + (z - 4)^2 = 49.

Solution: The given equation is in the general form. Comparing it with the general equation, we have:

• Center (h, k, l) = (2, -3, 4)
• Radius r = \sqrt{49} = 7

Example 3: Intersection with a Plane

Determine if the plane x + y + z = 10 intersects the sphere x^2 + y^2 + z^2 = 100.

Solution: To find the intersection, we can substitute z = 10 - x - y into the sphere's equation and solve for x and y. If there are real solutions, the plane intersects the sphere.

Substituting z into the sphere's equation:

[ x^2 + y^2 + (10 - x - y)^2 = 100 ]

Expanding and simplifying:

[ x^2 + y^2 + 100 - 20x - 20y + x^2 + y^2 + 2xy = 100 ]

[ 2x^2 + 2y^2 + 2xy - 20x - 20y = 0 ]

This is a quadratic equation in x and y, and it will have real solutions, indicating that the plane does intersect the sphere.

By understanding the equation of a sphere in Cartesian form, one can analyze various properties of the sphere, such as its size, position, and interactions with other geometric entities like planes and lines.