GEOMETRYCoordinate GeometryMathematics Calculator
●

Sphere Equation

A sphere with center (h,k,l) and radius r has equation (x−h)²+(y−k)²+(z−l)² = r². Volume V = (4/3)πr³, surface area A = 4πr². General form: x²+y²+z²+Dx+Ey+Fz+G = 0.

Concept Fundamentals
(x−h)²+(y−k)²+(z−l)² = r²
Standard
V = (4/3)πr³
Volume
A = 4πr²
Surface
d = 2r
Diameter

Did our AI summary help? Let us know.

All points at distance r from center. Circle equation extends to 3D as sphere. Completing square gives center and radius.

Key quantities
(x−h)²+(y−k)²+(z−l)² = r²
Standard
Key relation
V = (4/3)πr³
Volume
Key relation
A = 4πr²
Surface
Key relation
d = 2r
Diameter
Key relation

Ready to run the numbers?

Why: Sphere equations appear in physics (wave fronts, orbitals), computer graphics (bounding spheres, environment mapping), and geometry (distance from center).

How: Standard form: (x−h)²+(y−k)²+(z−l)² = r². Expand to get general form. Complete the square to go from general to standard. Volume and surface area use r only.

All points at distance r from center.Circle equation extends to 3D as sphere.
Sources:Khan Academy — SphereWolfram MathWorld — Sphere

Run the calculator when you are ready.

Find Sphere EquationEnter center and radius

Sphere Parameters

3D Visualization

Enter sphere parameters to see the 3D visualization

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

●

(x−h)²+(y−k)²+(z−l)² = r².

— Standard Form

V

V = (4/3)πr³, A = 4πr².

— Formulas

Key Takeaways

  • • A sphere is the set of all points in 3D equidistant from a center.
  • • Standard form: (x−h)2+(y−k)2+(z−l)2=r2(x-h)^2+(y-k)^2+(z-l)^2=r^2 with center (h,k,l) and radius r.
  • • General form: x2+y2+z2+Dx+Ey+Fz+G=0x^2+y^2+z^2+Dx+Ey+Fz+G=0 where D=-2h, E=-2k, F=-2l, G=h²+k²+l²-r².
  • • Volume = frac43pir3\\frac{4}{3}\\pi r^3; surface area = 4pir24\\pi r^2.
  • • The diameter is 2r — the longest chord through the center.

Did You Know?

Bubbles

Soap bubbles form spheres because a sphere minimizes surface area for a given volume.

Planets

Planets are approximately spherical due to gravity pulling matter toward the center.

Volume Ratio

A sphere has the smallest surface area for its volume of any 3D shape.

Great Circles

Any plane through the center of a sphere cuts it in a "great circle" — the largest possible circle on the sphere.

Antipodal Points

Points on opposite sides of a sphere (through the center) are called antipodal points.

Spherical Coordinates

Spheres are natural in spherical coordinates: (r, θ, φ) where r is constant.

Understanding Sphere Equations

A sphere in 3D is defined by its center (h,k,l) and radius r. Every point (x,y,z) on the sphere satisfies the distance equation.

(x−h)2+(y−k)2+(z−l)2=r2(x-h)^2+(y-k)^2+(z-l)^2=r^2

Expanding gives the general form with D=-2h, E=-2k, F=-2l.

Expert Tips

Center from General Form

h = -D/2, k = -E/2, l = -F/2. Complete the square to convert general to standard.

Point on Sphere

Substitute (x,y,z) into the equation. If LHS = r², the point is on the sphere.

Sphere vs Ball

A sphere is the 2D surface; a ball includes the interior. Volume refers to the ball.

Four Points

Four non-coplanar points uniquely determine a sphere. Three points define infinitely many.

Frequently Asked Questions

What is the standard form of a sphere?

(x-h)² + (y-k)² + (z-l)² = r², where (h,k,l) is the center and r is the radius.

How do I find the center from general form?

Center is (-D/2, -E/2, -F/2) when the equation is x²+y²+z²+Dx+Ey+Fz+G=0.

What is the volume of a sphere?

V = (4/3)πr³. This is two-thirds of the volume of the circumscribed cylinder.

What is the surface area?

A = 4πr². The derivative of volume with respect to r equals the surface area.

Can a sphere have a negative radius?

No. Radius is a distance and must be positive.

How many points determine a sphere?

Four non-coplanar points uniquely determine a sphere. Three points lie on infinitely many spheres.

What is a hemisphere?

Half of a sphere. Its volume is (2/3)πr³ and curved surface area is 2πr².

How to Use This Calculator

  1. Enter center (x, y, z) and radius. Click a sample example to auto-fill.
  2. Results update automatically — standard form, general form, volume, surface area, diameter.
  3. View the 3D visualization; drag to rotate and scroll to zoom.
  4. Check the step-by-step solution for the derivation.
  5. Copy results to share or paste into assignments.

Note: Radius must be positive. The 3D visualization shows wireframe circles. Results are for educational use.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Circle Equation Calculator

Circle Equation Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Ellipse Equation Calculator

Ellipse Equation Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Parabola Equation Calculator

Parabola Equation Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Centroid Calculator

Centroid Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Endpoint Calculator

Endpoint Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Gradient Calculator

Gradient Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics