Free Equation of a Sphere Calculator

Overview of the Sphere Equation Tool

This calculator helps you generate the equation of a sphere in either standard form or expanded form when relevant parameters are known. It can also extract the sphere’s center and radius directly from its equation. Beyond that, the tool computes the sphere’s surface area and volume, making it a complete geometric assistant.

The instructions below cover all five input modes:

• Center and radius supplied
• Standard equation entered
• Expanded equation entered
• Endpoints of a diameter provided
• Center plus a point on the sphere known

Each mode returns the sphere equation, center, radius, and optionally the derived properties.

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Standard Form of the Sphere Equation

A sphere is defined as the set of all points at a fixed distance $r$ from a center $(h,k,l)$. The standard equation is:

$$(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$$

Where:

• $(x,y,z)$ – coordinates of any point on the sphere’s surface
• $(h,k,l)$ – coordinates of the center
• $r$ – radius

Example: from center and radius

If the center is $(3,7,5)$ and $r = 10$, substitute directly:

$$(x-3)^2 + (y-7)^2 + (z-5)^2 = 100$$

Conversely, given the standard equation you can read off the center and radius. For instance:

$$(x-7)^2 + (y-12)^2 + (z-4)^2 = 36$$

has center $(7,12,4)$ and radius $\sqrt{36}=6$. Pay attention to signs: if the equation were $(x-7)^2 + (y+12)^2 + (z+4)^2 = 36$, the center becomes $(7,-12,-4)$.

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How the Standard Equation Is Derived

Take a point $S(x,y,z)$ on the sphere surface at distance $r$ from center $(h,k,l)$. By the three‑dimensional distance formula:

$$\sqrt{(x-h)^2 + (y-k)^2 + (z-l)^2} = r$$

Squaring both sides yields the standard form.

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Expanded Form of the Sphere Equation

Occasionally you encounter a sphere equation that looks different:

$$x^2 + y^2 + z^2 + Ex + Fy + Gz + H = 0$$

where $E,F,G$ are coefficients of the linear terms and $H$ is the constant. This is simply the expanded version of the standard form. Using the completing‑the‑square method, you can rewrite it back to standard form.

Conversion steps

1. Group like terms:
   $(x^2 + Ex) + (y^2 + Fy) + (z^2 + Gz) = -H$

2. Complete each square: for $x$-terms, add $\left(\dfrac{E}{2}\right)^2$ and subtract it; similarly for $y$ and $z$.

3. Rearrange to isolate the squared terms:

$$\left(x + \dfrac{E}{2}\right)^2 + \left(y + \dfrac{F}{2}\right)^2 + \left(z + \dfrac{G}{2}\right)^2 = \left(\dfrac{E}{2}\right)^2 + \left(\dfrac{F}{2}\right)^2 + \left(\dfrac{G}{2}\right)^2 - H$$

4. Compare with the standard form to extract center and radius:

\begin{aligned} h &= -\dfrac{E}{2}, \quad k = -\dfrac{F}{2}, \quad l = -\dfrac{G}{2} \$$4pt] r &= \sqrt{ \left(\dfrac{E}{2}\right)^2 + \left(\dfrac{F}{2}\right)^2 + \left(\dfrac{G}{2}\right)^2 - H } \end{aligned}

The calculator automates this conversion when you supply $E,F,G,H$.

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Equations from Diameter Endpoints or a Center+Point

Diameter endpoints

Let the endpoints of any diameter be $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$. The center is the midpoint:

$$(h,k,l) = \left(\dfrac{x_1+x_2}{2},\; \dfrac{y_1+y_2}{2},\; \dfrac{z_1+z_2}{2}\right)$$

The radius is half the distance between $A$ and $B$:

$$r = \dfrac{1}{2}\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$

Substituting $h,k,l,r$ into the standard form gives the sphere’s equation.

Center and a known point on the surface

If the center $(h,k,l)$ and a point $P(p_x,p_y,p_z)$ on the sphere are known, compute the radius via:

$$r = \sqrt{(p_x-h)^2 + (p_y-k)^2 + (p_z-l)^2}$$

Then use the standard form with the known center and this radius.

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Using This Calculator

The tool offers five input modes, selectable from a drop‑down:

1. Center & radius – directly enter $(h,k,l)$ and $r$. The standard and expanded equations are returned.
2. Standard equation – provide $h,k,l$ and $r^2$. The calculator displays the equation and derives center and radius.
3. Expanded equation – enter $E,F,G,H$. The tool converts to standard form, shows the center and radius.
4. Diameter endpoints – input coordinates of two end‑points. The calculator computes the midpoint (center), half‑distance (radius), and the full equation.
5. Center + point on sphere – give the center and a surface point. The radius is found, then the equation is generated.

After obtaining the radius, the calculator also computes the sphere’s surface area ($4\pi r^2$) and volume ($\frac{4}{3}\pi r^3$). These results appear alongside the equation.

If the tool ignores inputs in a particular mode, refresh the calculator using the circular reload icon at the bottom and re‑enter the values.

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Key Points to Remember

• The standard form $(x-h)^2+(y-k)^2+(z-l)^2 = r^2$ is the most direct way to describe a sphere.
• The expanded form $x^2+y^2+z^2+Ex+Fy+Gz+H=0$ can always be transformed back via completing the square.
• The distance formula is the foundation for deriving the equation from center/radius, diameter endpoints, or a center+point.
• This calculator supports all commonly used input scenarios, making it a versatile tool for finding the center and radius of a sphere and for writing the sphere equation in standard form or expanded form on the fly.