Equation of a Sphere Calculator

How to use the equation of a sphere calculator

What this calculator does

This calculator turns several common descriptions of a sphere into one consistent model. It finds the standard equation, expanded equation, center, radius, diameter, surface area, and volume. It can start from a center and radius, standard-form coefficients, expanded-form coefficients, two endpoints of a diameter, or a known center plus one surface point. The result is an exact geometric identity apart from displayed decimal rounding; it does not determine whether measured real-world data form a perfect sphere.

When to use it

Use it when checking analytic-geometry homework, converting a quadratic equation into center-radius form, constructing a sphere from 3D coordinates, or estimating area and volume after the radius is known. The underlying equation follows directly from the three-dimensional distance formula; the OpenStax-based guide to vectors in three dimensions gives additional coordinate-system context.

How to calculate

1. The calculator opens with a complete demonstration: center (3, 7, 5) and radius 10. Its results and a validated example XLSX workbook are available immediately.
2. Choose a description in Known information. Replace the active example values with your own signed decimal values. Use a period as the decimal separator; commas and scientific notation are intentionally rejected to prevent ambiguous input.
3. Read the Standard equation first, then use the result cards and coefficient table to verify the center, radius, expanded coefficients, surface area, and volume.
4. Select Download Excel to export the current canonical values and equations. Reset clears all numeric fields and results; Excel export stays disabled until a complete valid set is entered again.

Input guide

Known information is required and selects which field set is active. In Center and radius mode, Center h, Center k, and Center l are any finite real coordinates, while Radius r must be positive; for example, h = 3, k = 7, l = 5, and r = 10. A larger radius increases diameter linearly, surface area quadratically, and volume cubically.

In Standard equation coefficients mode, enter h, k, l, and R = r² from (x – h)² + (y – k)² + (z – l)² = R. R must be positive. Watch the signs: a written factor (y + 4)² means k = – 4. In Expanded equation coefficients mode, enter E, F, G, and H from x² + y² + z² + Ex + Fy + Gz + H = 0. These values must imply a positive radius squared; otherwise the equation does not describe a real non-degenerate sphere.

In Endpoints of a diameter mode, Point A coordinates x₁, y₁, z₁ and Point B coordinates x₂, y₂, z₂ are all required. The two points must differ. The center is their midpoint and the radius is half their distance. In Center and a point on the sphere mode, Center h, k, l and Surface point pₓ, pᵧ, p_z are required. The surface point must not equal the center because that would give a zero-radius degenerate case.

Output guide

Standard equation is the primary exact identity. Expanded equation shows the same sphere after algebraic expansion. Center is the ordered triple (h, k, l). Radius is the center-to-surface distance, Diameter is 2r, and Radius squared is the standard equation's right-hand constant. Surface area is 4πr² in square units, while Volume is (4/3)πr³ in cubic units. The summary pills repeat the active mode, center, radius, and volume; the coefficient table reports h, k, l, E, F, G, and H from the same model.

Worked example

For the startup values h = 3, k = 7, l = 5, and r = 10, radius squared is 100. Substitution gives (x – 3)² + (y – 7)² + (z – 5)² = 100. Expanding produces E = – 6, F = – 14, G = – 10, and H = 3² + 7² + 5² – 10² = – 17, so the expanded equation is x² + y² + z² – 6x – 14y – 10z – 17 = 0. The diameter is 20, surface area is approximately 1,256.637 square units, and volume is approximately 4,188.790 cubic units.