Equation of a Sphere Calculator – Guide & Formulas
Find the equation of a sphere given center and radius, or four points. Free online calculator with step-by-step solutions for standard and general sphere equations.
Find the equation of a sphere in 3D space. Enter the center and radius, or four non-coplanar points, to instantly get the sphere equation with step-by-step explanations.
Key Takeaway
Use the free Equation of a Sphere Calculator to find the equation of a sphere given center and radius, or four points. free online calculator with step-by-step solutions for standard and general sphere equations. Get instant results with step-by-step explanations.
How to Use the Equation of a Sphere Calculator
- Select input mode: Center + Radius or Four Points.
- Enter the center coordinates and radius, or four point coordinates.
- Click Calculate to find the sphere equation in standard and general form.
- Review the center, radius, volume, surface area, and step-by-step derivation.
The Formula
Variable Definitions
- (h, k, l): Center of the sphere
- r: Radius of the sphere
- (x, y, z): Any point on the sphere surface
- D, E, F, G: Coefficients in the general form equation
Sphere with Center (2, 3, 4) and Radius 5
Find the equation of a sphere given its center and radius.
- Step 1: Center = (2, 3, 4), Radius = 5.
- Step 2: Standard form: (x − 2)² + (y − 3)² + (z − 4)² = 25.
- Step 3: Expand: x² − 4x + 4 + y² − 6y + 9 + z² − 8z + 16 = 25.
- Step 4: General form: x² + y² + z² − 4x − 6y − 8z + 4 = 0.
- Step 5: Volume = (4/3)π(5³) = 523.6 cubic units. Surface Area = 4π(5²) = 314.16 square units.
Frequently Asked Questions
What is a sphere?
A sphere is the set of all points in 3D space that are equidistant from a fixed center point. That constant distance is the radius. A sphere is the 3D analog of a circle.
How do four points define a sphere?
Four non-coplanar points uniquely determine a sphere. The center is equidistant from all four points, leading to a system of equations that solves for the center and radius.
What is the general form of a sphere?
The general form is x² + y² + z² + Dx + Ey + Fz + G = 0. The center is (−D/2, −E/2, −F/2) and the radius is √((D/2)² + (E/2)² + (F/2)² − G).
When does a sphere equation have no real solution?
If the radius squared is negative (r² < 0), the equation has no real solution. This happens when the four points are coplanar or when the general form coefficients are inconsistent.
How do I find the volume of a sphere?
Volume = (4/3)πr³. For the example: (4/3)π(5³) = 523.6 cubic units.
How do I find the surface area of a sphere?
Surface Area = 4πr². For the example: 4π(5²) = 314.16 square units.
What is a great circle?
A great circle is the intersection of a sphere with a plane passing through its center. It is the largest possible circle on the sphere and divides it into two equal hemispheres.
Can a sphere be degenerate?
Yes. If the radius is zero, the sphere collapses to a single point (the center). If the radius is imaginary, no real sphere exists.
How is a sphere different from a circle?
A circle is a 2D curve — all points at distance r from a center in a plane. A sphere is a 3D surface — all points at distance r from a center in space.
What are real-world applications of sphere equations?
Sphere equations are used in GPS calculations, 3D graphics, ball bearings design, planetary science, medical imaging (CT scans), and acoustic modeling.