Derivation of Equation of Sphere

To derive the equation of a sphere, let’s start with a sphere centered at (h, k, l) with radius r. Now, consider any point (x, y, z) on the sphere’s surface.

The distance between this point and the sphere’s center, according to the three-dimensional distance formula, is:

Distance = √((x – h)² + (y – k)² + (z – l)²)

For this point to be on the sphere’s surface, this distance must equal the radius r: √((x – h)² + (y – k)² + (z – l)²) = r

Squaring both sides to eliminate the square root gives us: (x – h)² + (y – k)² + (z – l)² = r²

This equation defines all points in space equidistant from the center, hence describing the surface of the sphere.

Example: For a sphere centered at (3, −2, 1) with a radius of 5.

Solution:

Equation of sphere centered at (h, k, l) with radius r: (x – h)² + (y – k)² + (z – l)² = r²

Thus, equation of sphere centered at (3, -2, 1) is:

(x − 3)²+ (y + 2)²+ (z − 1)² = 5²

This equation represents all points (x, y, z) that are 5 units away from the point (3, −2, 1).

Equation of a Sphere

The equation of a sphere defines all points equidistant from its center, given by (x – h)² + (y – k)² + (z – l)² = r², where (h, k, l) is the center and r is the radius. This article provides an in-depth exploration of the equation of a sphere, its properties, applications, and related concepts.

Table of Content

What is Sphere?
Equation of Sphere

General Equation of Sphere
Parametric Equations of a Sphere
Geometrical Interpretation of the Equation of a Sphere