Equation of a Sphere Calculator
Find sphere equations, center coordinates, radius, and geometric properties
Calculate Sphere Equation
Enter Center Coordinates and Radius
Enter Values
Please enter valid values to calculate sphere properties.
Example Calculation
Given Center and Radius
Center: (3, 7, 5)
Radius: 10
Standard Form:
(x-3)² + (y-7)² + (z-5)² = 100
Surface Area: 4π(10)² = 400π ≈ 1256.6
Volume: (4/3)π(10)³ = 4000π/3 ≈ 4188.8
Sphere Equation Forms
Standard Form
(x-h)² + (y-k)² + (z-l)² = r²
Center: (h,k,l), Radius: r
Expanded Form
x² + y² + z² + Ex + Fy + Gz + H = 0
Center: (-E/2, -F/2, -G/2)
Quick Tips
A sphere is the set of all points equidistant from a center point
The center coordinates determine sphere position in 3D space
Radius must be positive for a valid sphere
Surface area = 4πr², Volume = (4/3)πr³
Understanding Sphere Equations
What is a Sphere Equation?
The equation of a sphere represents all points in 3D space that are equidistant from a center point. The standard form clearly shows the center coordinates and radius, making it easy to visualize the sphere's position and size.
Standard Form
(x - h)² + (y - k)² + (z - l)² = r²
- (h, k, l): Center coordinates
- r: Radius of the sphere
- (x, y, z): Any point on the sphere
Expanded Form
x² + y² + z² + Ex + Fy + Gz + H = 0
The expanded form is obtained by expanding the standard form. To convert back to standard form, we use the completing the square method:
- Center: (-E/2, -F/2, -G/2)
- Radius: √((E² + F² + G²)/4 - H)
Note: For a valid sphere, (E² + F² + G²)/4 - H must be positive.
From Diameter Endpoints
If you know the endpoints of a diameter, you can find the sphere equation:
- 1. Find the center as the midpoint of the diameter
- 2. Calculate radius as half the diameter length
- 3. Use the center and radius in standard form
From Center and Point
If you know the center and any point on the sphere:
- 1. Use the distance formula to find radius
- 2. r = √[(x₁-h)² + (y₁-k)² + (z₁-l)²]
- 3. Substitute into standard form equation