Equation of a Sphere

A sphere is a 3 – dimensional object. It is the collection of all the points equidistant from a fixed point, the center. It can be referred to as the 3-d version of a circle. So its equation is almost similar to that of a circle with an added variable for the extra dimension.

In Standard Form

The standard equation of a sphere given the center and radius is:

(x – h)² + (y – k)² + (z – l)² = r², here (x, y, z) = any point on the sphere, (h, k, l) = center, r = radius

When the center is at the origin (0, 0, 0), the equation of a sphere in standard form becomes:

(x – 0)² + (y – 0)² + (z – 0)² = r²

⇒ x² + y² + z² = r²

Let us derive the equation.

Derivation

Let (h, k, l) be a fixed point in the 3 – D plane.

Let (x, y, z) be another point from (h, k, l) at a distance r such that r is constant and a real positive number.

Squaring both sides,

(x – h)² + (y – k)² + (z – l)² = r² (applying the Pythagorean theorem)

So the equation of sphere with center and radius is:

(x – h)² + (y – k)² + (z – l)² = r²

r = √[(x – h)² + (y – k)² + (z – l)² ] ——— (applying the distance formula)

If the center is at the origin (0, 0, 0)

Then the equation of the sphere is:

x² + y² + z² = r²

∴r = √( x² + y² + z²)

Let us solve some examples to understand the above concept.

In General Form

The equation of a sphere in general (expanded) form is written as:

x² + y² + z² + 2ux + 2vy + 2wz + d = 0, here (-u, -v, -w) = center, and u, v, w, & d = constants

∵ Radius (r) = ${\sqrt{u^{2}+v^{2}+w^{2}-d}}$, here (-u, -v, -w) = center, and u, v, w, & d = constants, then

d = u² + v² + w² – r²

Let us solve some examples to understand the above concept.