Sphere: Equation, Center-Radius Form and Key Properties

A sphere is the set of all points in 3D at a fixed distance r, the radius, from a fixed point (a, b, c), the center. Its equation is (x − a)² + (y − b)² + (z − c)² = r² — the 3D version of the circle equation.

Expanding gives the general form x² + y² + z² + 2ux + 2vy + 2wz + d = 0, from which the center (−u, −v, −w) and radius √(u² + v² + w² − d) can be read off by completing the square.

Key Concepts

• Center–radius form: (x − a)² + (y − b)² + (z − c)² = r². Every point (x, y, z) on the sphere is exactly r units from (a, b, c).
• General form: x² + y² + z² + 2ux + 2vy + 2wz + d = 0. The coefficients of x², y², z² must be equal and non-zero for the equation to describe a sphere.
• Center = (−u, −v, −w); radius = √(u² + v² + w² − d). A real sphere needs u² + v² + w² − d > 0.
• If u² + v² + w² − d = 0, the "sphere" is a single point; if it is negative, no real points satisfy the equation.
• Diameter form: if the diameter has endpoints P(x1, y1, z1) and Q(x2, y2, z2), the sphere is (x − x1)(x − x2) + (y − y1)(y − y2) + (z − z1)(z − z2) = 0.
• A plane section of a sphere is a circle; a plane through the center cuts a great circle of radius r.

Formulas

• (x − a)² + (y − b)² + (z − c)² = r².
• General: x² + y² + z² + 2ux + 2vy + 2wz + d = 0; center (−u, −v, −w); radius √(u² + v² + w² − d).
• Diameter form: (x − x1)(x − x2) + (y − y1)(y − y2) + (z − z1)(z − z2) = 0.
• A point P at distance dist from the center is inside the sphere if dist < r, on it if dist = r, and outside if dist > r.

Worked Example

Find the center and radius of x² + y² + z² − 4x + 6y − 8z = 11.

1. Match to the general form: 2u = −4 gives u = −2; 2v = 6 gives v = 3; 2w = −8 gives w = −4; d = −11.
2. Center = (−u, −v, −w) = (2, −3, 4).
3. Radius = √(u² + v² + w² − d) = √(4 + 9 + 16 + 11) = √40 = 2√10 ≈ 6.32.
4. Check by completing the square: (x − 2)² + (y + 3)² + (z − 4)² = 11 + 4 + 9 + 16 = 40. Matches.

Common Misconceptions

• Any equation with x², y², z² is a sphere. The squared coefficients must be equal, and there must be no xy, yz or zx cross terms.
• The center is (u, v, w). It is (−u, −v, −w) — the signs flip. Forgetting this is the most common error.
• The radius is √(u² + v² + w² + d). The formula subtracts d: √(u² + v² + w² − d). With d = −11 above, that gives √40, not √18.

Real-World Applications

• GPS: position is found from spheres centered on satellites, a technique called trilateration.
• Computer graphics and games: bounding spheres for fast collision checks.
• Astronomy: celestial-sphere models and planetary surfaces.
• Chemistry: atomic and molecular models represent atoms as spheres.

A sphere is (x − a)² + (y − b)² + (z − c)² = r², or in general form with center (−u, −v, −w) and radius √(u² + v² + w² − d). Watch the sign flips, check the radius condition, and complete the square to verify.