Sphere Equation Calculator - Center, Radius, Volume

A sphere equation calculator is an analytic-geometry tool that turns a sphere's standard or general form into its center, radius, surface area, and volume. Given (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 or x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0, the page shows the recovered center, the radius, and both forms of the equation.

• • Convert between standard and general form: Move from one form to the other without completing the square by hand.
• • Recover the center and radius: Read the center (h, k, l) and the radius r directly from the coefficients.
• • Check homework or exam answers: Verify the center and radius match the original general-form equation.

The page also flags when the equation does not actually describe a sphere of positive radius, so a sign error shows up the moment you finish typing.

If your equation is the 2D version with just x and y, the Circle Equation Calculator applies the same standard-to-general conversion in two coordinates.

The calculator reads the parameters of whichever form you choose, then uses the identity that expands (x - h)^2 + (y - k)^2 + (z - l)^2 to recover the center, the radius squared, and the parameters of the other form.

• h, k, l: Coordinates of the sphere's center in the standard form.
• r^2: Right-hand side of the standard form, equal to the squared radius.
• D, E, F: Linear coefficients of x, y, z in the general form.
• G: Constant term in the general form. Drives the squared radius through (D/2)^2 + (E/2)^2 + (F/2)^2 - G.

The two forms are linked by expanding the standard form: x^2 + y^2 + z^2 - 2h x - 2k y - 2l z + h^2 + k^2 + l^2 - r^2 = 0, so D = -2h, E = -2k, F = -2l, and G = h^2 + k^2 + l^2 - r^2. Once the center and the radius are known, surface area is 4 pi r^2 and volume is (4/3) pi r^3.

According to Wolfram MathWorld, a sphere in standard form is (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 with center (h, k, l) and radius r, and the same sphere in general form is x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0 with center (-D/2, -E/2, -F/2).

As published by OpenStax Calculus Volume 3, the standard form of a sphere with center (h, k, l) and radius r is (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, and the corresponding general form is x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0.

Once the center and radius are known, the Sphere Calculator takes a radius or diameter and reports the volume, surface area, and A/V ratio directly without retyping the equation.

Four ideas show up every time a sphere is described by an equation.

The standard form is a squared version of the 3D distance formula: the distance from (x, y, z) to (h, k, l) equals the radius. The general form is the same equation with the squares expanded, so the linear coefficients are -2h, -2k, and -2l.

The standard form of a sphere is the 3D distance formula squared, so the 3D Distance Calculator is the natural companion when a problem is phrased around the distance from a point to the center.

Pick the form you have, type the four parameters, and read the center, radius, and the alternate equation from the results panel.

1. 1 Choose the input form: Use the dropdown to pick Standard (h, k, l, r^2) or General (D, E, F, G). Only the matching group of fields is used in the calculation.
2. 2 Enter the standard-form parameters: Type the center x, y, z coordinates and the squared radius r^2. The defaults give the example sphere (1, -2, 3) with r^2 = 16.
3. 3 Enter the general-form parameters: Type D, E, F, and G from x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0. The defaults (D = -2, E = 4, F = -6, G = -11) match the same example.
4. 4 Read the center and radius: The primary output is the center (h, k, l). The radius, diameter, r^2, surface area, and volume refresh on every keystroke below it.
5. 5 Read the alternate equation: The result panel shows the equation written in the form you did not start from, so you can paste it into notes or a 3D plot.

Try the general-form inputs D = -2, E = 4, F = -6, G = -11. The calculator returns center (1, -2, 3), radius 5, surface area 314.16, and volume 523.60. Switch to Standard and the same page shows (x - 1)^2 + (y + 2)^2 + (z - 3)^2 = 25.

If you would rather skip the equation and enter a radius directly, the Sphere Volume Calculator returns the volume and surface area in one step.

The benefits matter most when you are working a 3D geometry problem by hand and need a quick check on the center, radius, and the second form of the equation.

• • Skip the completing-the-square algebra: Manual sphere problems are easy to slip on when completing the square in three coordinates at once. The calculator handles the algebra so you can focus on the rest of the problem.
• • See center, radius, and the other form in one pass: Type the parameters once and the page shows the center, the radius, the surface area, the volume, and the equation in the form you did not start from.
• • Catch sign errors with the r^2 check: The status line flags 'No real sphere' the moment the inputs produce a negative r^2, so a sign slip is visible before you submit your work.
• • Work in standard or general form interchangeably: If your textbook starts in general form and your sketch is in standard form, the form dropdown keeps both views a click away without retyping.

The page is also a good way to build intuition about how D, E, F, and G relate to the center coordinates. Watching h = -D/2 and k = -E/2 line up is a faster route to memorizing the identities than working through them by hand.

If the next step is the half-sphere version of the same shape, the Area of Hemisphere Calculator reports the curved and total surface area plus the hemisphere volume from the radius you just recovered.

The formula is the same in every case, but a few factors change how the recovered center and radius should be read.

• • This page is the 3D sphere case only. For a 2D circle of the same form, the circle-equation calculator handles the two-coordinate version of the same algebra.
• • The calculator assumes a Euclidean sphere. It does not describe an oblate spheroid, a parametric sphere, or any surface where the leading coefficients on x^2, y^2, and z^2 are not all equal.
• • Negative r^2 is a valid algebraic result, not an input error. The page reports 'No real sphere' so the sign is visible, but it cannot plot an imaginary radius.

If the coefficients on x^2, y^2, and z^2 are not all the same sign, the equation is an ellipsoid, hyperboloid, or paraboloid, not a sphere, and the page should not be used. Divide through first if the leading coefficients are not all 1.

According to Wikipedia (Sphere), the surface area of a sphere of radius r is 4 pi r^2 and the volume is (4/3) pi r^3, and the general-form equation x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0 has center at (-D/2, -E/2, -F/2) with r^2 = (D/2)^2 + (E/2)^2 + (F/2)^2 - G.

When the problem is phrased around the half-ball, the Volume of Hemisphere Calculator uses the same radius to compute the hemisphere volume without re-entering coefficients.