Equation of a Sphere Calculator
Find standard and expanded sphere coefficients from its center and radius.
Formula
(x−h)²+(y−k)²+(z−l)²=r²; x²+y²+z²+Dx+Ey+Fz+G=0; D=−2h; E=−2k; F=−2l; G=h²+k²+l²−r².
r² ≈ (4)^2 · D ≈ -2×(1) · E ≈ -2×(2) · F ≈ -2×(3) · G ≈ (1)^2+(2)^2+(3)^2-(16)
How to use
- Enter center h,k,l.
- Enter the positive radius in the same unit.
- Use r² or D,E,F,G in the displayed equation.
Examples
- Center (1,2,3), radius 4 gives D=−2,E=−4,F=−6,G=−2.
Frequently asked questions
How do I use the expanded form?
Substitute the four output coefficients into x²+y²+z²+Dx+Ey+Fz+G=0.
What if I know diameter endpoints?
Their midpoint is the center; half their distance is the radius.
Do the coefficients need units?
Keep one coordinate unit throughout; r² and G use its square.
Sources
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