⭕ Equation of a Sphere Calculator
Calculate sphere equation from center and radius
Center (h, k, l)
Radius must be positive
How to Use This Calculator
Enter Center Coordinates
Input the x, y, and z coordinates (h, k, l) of the sphere's center point.
Enter Radius
Input the radius r of the sphere (must be positive).
Click Calculate
Press "Calculate Equation" to get both standard and expanded forms of the sphere equation.
Formula
Standard Form: (x - h)² + (y - k)² + (z - l)² = r²
Expanded Form: x² + y² + z² - 2hx - 2ky - 2lz + (h² + k² + l² - r²) = 0
Where:
- (h, k, l) = Center of the sphere
- r = Radius of the sphere (r > 0)
- (x, y, z) = Any point on the sphere
Example: Center (0, 0, 0), Radius 5
Standard: (x - 0)² + (y - 0)² + (z - 0)² = 5²
Standard: x² + y² + z² = 25
Expanded: x² + y² + z² - 0x - 0y - 0z + (0 - 25) = 0
Expanded: x² + y² + z² - 25 = 0
About Equation of a Sphere Calculator
The Equation of a Sphere Calculator finds the mathematical equation of a sphere given its center coordinates and radius. It provides both standard form and expanded form of the equation.
When to Use This Calculator
- Calculus: Set up integrals over spherical regions
- Geometry: Analyze spheres in 3D coordinate space
- Physics: Model spherical objects, particles, or fields
- Engineering: Design spherical components and structures
- Computer Graphics: Render spheres and spherical surfaces
- Mathematics: Solve problems involving spheres
Why Use Our Calculator?
- ✅ Dual Forms: Provides both standard and expanded forms
- ✅ Step-by-Step Display: Shows the conversion process
- ✅ Works with All Numbers: Handles decimals, fractions, and negatives
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Educational: Helps understand sphere equations
- ✅ Completely Free: No registration required
Understanding Sphere Equations
A sphere is the set of all points equidistant from a fixed point (the center). The distance from any point on the sphere to the center equals the radius.
- Standard form: (x - h)² + (y - k)² + (z - l)² = r²
- Center at origin: x² + y² + z² = r²
- All points (x, y, z) satisfying the equation lie on the sphere
- Radius r must be positive
- Sphere is 3D analog of a circle (2D)
Real-World Applications
Physics: Spheres model planets, atoms, bubbles, and other spherical objects. The equation helps calculate volumes, surface areas, and gravitational fields.
Engineering: Spherical bearings, tanks, and domes use sphere equations for design and analysis.
Computer Graphics: 3D rendering uses sphere equations to render balls, planets, and other spherical objects efficiently.
Frequently Asked Questions
What is the equation of a sphere?
The standard form is (x - h)² + (y - k)² + (z - l)² = r², where (h, k, l) is the center and r is the radius. It describes all points equidistant from the center.
How do I convert standard form to expanded form?
Expand the squares: (x - h)² = x² - 2hx + h², etc. Then combine like terms. The expanded form is: x² + y² + z² - 2hx - 2ky - 2lz + (h² + k² + l² - r²) = 0.
What if the sphere is centered at the origin?
If center is (0, 0, 0), the equation simplifies to x² + y² + z² = r². This is the simplest form, like x² + y² = r² for a circle in 2D.
Can the radius be negative?
No! Radius must be positive (r > 0). A negative radius doesn't make geometric sense. The calculator requires a positive radius.
What's the difference from a circle equation?
Circle: (x - h)² + (y - k)² = r² (2D). Sphere: (x - h)² + (y - k)² + (z - l)² = r² (3D). Sphere adds a z-coordinate dimension.
How do I check if a point is on the sphere?
Substitute the point's coordinates into the sphere equation. If the left side equals r², the point is on the sphere. Otherwise, it's inside (less than) or outside (greater than).
What's the volume and surface area of a sphere?
Volume: V = (4/3)πr³. Surface area: A = 4πr². These are different from the sphere equation, which describes the shape, not volume/area.