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Last updated: July 31, 2026

Equation of a Sphere Calculator

Dharmendra Singh - Creator

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Balram Rajput
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Formula

standard: (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2; general: x^2 + y^2 + z^2 + d*x + e*y + f*z + g = 0

Where:

  • h=h
  • k=k
  • l=l
  • r=r
Equation of a SphereA sphere centered at (h, k, l) with radius r, shown with perspective ellipses to suggest three dimensions.(h, k, l)rStandard Form(x-h)²+(y-k)²+(z-l)²= r²Center (h,k,l), radius r
Every point on the sphere's surface is exactly r units from the center (h, k, l).

Worked Examples

Center (1,-2,3), radius 4

Generate both equation forms.

  1. 1d=-2, e=4, f=-6
  2. 2g=1+4+9-16=-2
Final Answer: generalForm = x² + y² + z² + -2x + 4y + -6z + -2 = 0

Origin-centered sphere

h=k=l=0.

  1. 1standard: x²+y²+z²=25
  2. 2g=-25
Final Answer: generalForm = x² + y² + z² + 0x + 0y + 0z + -25 = 0

Decimal center values

Use non-integer center and radius.

  1. 1Compute d,e,f,g
  2. 2Render standard/general forms
Final Answer: Both forms returned with rounded coefficients

Introduction

Convert a sphere center and radius into both standard and expanded general equations.

Formula and Method

Equation of a Sphere Calculator uses: standard: (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2; general: x^2 + y^2 + z^2 + d*x + e*y + f*z + g = 0

  • Enter all required inputs

  • Apply the underlying math formula(s)

  • Compute supporting terms where needed

  • Round numeric outputs for stable display

Input Fields

Provide values for: h, k, l, r.

  1. 1

    h: h

  2. 2

    k: k

  3. 3

    l: l

  4. 4

    r: r

Output Fields

The calculator returns: standardForm, d, e, f, g, generalForm.

  1. 1

    standardForm: Sphere equation in standard form

  2. 2

    d: General-form x coefficient

  3. 3

    e: General-form y coefficient

  4. 4

    f: General-form z coefficient

  5. 5

    g: General-form constant term

  6. 6

    generalForm: Sphere equation in general form

Validation Rules

Input validation is applied before calculation to prevent invalid math states.

  • h, k, l, r must be finite numbers.

  • r must be positive.

  • d, e, f, g are computed directly from center/radius.

  • Equation strings use rounded numeric values.

How to Use This Calculator

Fill in the inputs, run the calculator, and review primary output first.

  • Enter the required values

  • Click calculate

  • Check the primary output

  • Use example scenarios for verification

Practical Uses

This calculator supports learning, quick checks, and day-to-day technical problem solving.

  • Homework and exam preparation

  • Engineering and science sanity checks

  • Classroom demonstrations

  • Fast recalculation with different inputs

FAQs

What does the Equation of a Sphere Calculator compute?

Convert a sphere center and radius into both standard and expanded general equations.

Which inputs are required?

Required inputs: h, k, l, r.

How should I format numbers?

Use finite numeric values; decimals and negative numbers are allowed unless restricted by validation rules.

How are results rounded?

Numeric outputs are rounded to about 6 decimal places for readability.

Why might I see an error?

h, k, l, r must be finite numbers.

Can I use this for learning and verification?

Yes. Use the examples and formula section to verify manual calculations quickly.

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