# Equation of a Sphere Calculator

Our **equation of a sphere** calculator will help you write the equation of a sphere in the **standard form** or **expanded form** if you know the **center** and **radius** of the sphere. Alternatively, you can find the sphere equation if you know its center and **any point on its surface** or if you know the **end-points** of any of its **diameters**. This calculator can also find the center and radius of a sphere if you know its equation.

Are you wondering what is the standard equation of a sphere or how to **find the center and radius** of a sphere using its equation? Have you perhaps come across a sphere equation that doesn't look like the general equation? Are you curious how knowing the **diameter's end-points** or the center and a **point on the sphere** can help you **find the sphere's equation?** Grab your favorite drink and keep reading this article, and we'll tackle these questions together!

## What is the equation of a sphere?

The equation of a sphere in the standard form is given by:

where:

- $(x,y,z)$ – Coordinates of
**any point lying on the surface**of the sphere; - $(h,k,l)$ – Coordinates of the
**center**of the sphere; and - $r$ –
**Radius**of the sphere.

If we know the **center** and **radius** of the sphere, we can plug them into this standard form to obtain the **equation of the sphere**. For example, consider a sphere with a **radius** of $10$ and its **center** at $(3,7,5)$. By inserting this into the equation above, we get the standard equation of the sphere:

Similarly, you can use the standard form of the sphere equation to **find the radius and center** of the sphere. For example, a sphere with the equation $(x-7)^2 + (y-12)^2 + (z-4)^2 = 36$ would have its center at $(7,12,4)$, and its radius is given by:

❗ Note that you have to be careful regarding the signs of the center coordinates in the standard equation. If the sphere equation in the above example was instead $\small(x-7)^2 + (y+12)^2 + (z+4)^2 = 36$, then its center would be $\small(7,-12,-4)$.

## How to derive the equation of a sphere

Now that we know the standard equation of a sphere, let's learn how it came to be:

- First thing to understand is that the equation of a sphere represents
**all the points lying equidistant from a center**point. In other words, any point that lies at a distance $r$ from a point $(h,k,l)$ lies on the sphere. This concept is similar to how the equation of a circle works. - Consider a point $S(x,y,z)$ that lies at a distance $r$ from the center $(h,k,l)$. Using distance formula, we get:

- Squaring on both sides and rearranging this equation, we get:

And, voila! Just like that, we got the **standard equation of a sphere!** It bears repeating that every point $S$ that satisfies this equation lies on the surface of the sphere.

## Expanded form of the sphere equation

Sometimes, you may come across a sphere equation that appears **different from the standard** form we've discussed so far. Generally, such equations should look like the following:

where:

- $E$ – Sum of all coefficients of the $x$ terms;
- $F$ – Sum of all coefficients of the $y$ terms;
- $G$ – Sum of all coefficients of the $z$ terms; and
- $H$ – Sum of all constants terms.

This equation may seem intimidating at a glance, but if you take a closer look; it is simply the **expanded version of the standard form** we're used to. Let's use completing the squares method to backtrack our way from the expanded version to the simpler, standard form:

**Rearrange**the equation for clarity:

- To write $(x^2+Ex)$ in the $(a-b)^2 = a^2-2ab+b^2$ format, we need to identify $a$ and $b$. Clearly, $a=x$. If $-2ab = Ex$, we get:

**Adding**and**subtracting**the quantity $\textcolor{red}{\left(- \frac{E}{2} \right)^2}$ to the equation, we get:

- Similarly, we
**complete the squares**for the**y-terms**and**z-terms**by**adding**and**subtracting**the quantities $\textcolor{red}{\left(- \frac{F}{2} \right)^2}$ and $\textcolor{red}{\left(- \frac{G}{2} \right)^2}$ to get:

**Rearrange**the equation to group**all**the**constant terms**on the**right-hand side**:

**Comparing**this equation with the standard form of the sphere equation $(x-h)^2 +(y-k)^2 + (z-l)^2) = r^2\text{:}$

- Extracting the
**center**point and**radius**of the sphere from this equation, we get:

In this sphere equation calculator, you can simply select the expanded form option as the type of the equation and enter the corresponding $E$, $F$, $G$ and $H$ values to obtain the equation in the standard form, its center and radius, and other relevant information.

## Equation of a sphere from the end-points of any diameter

Now that you know what the equation of a sphere is, let's discuss how to obtain it in cases where you don't have the necessary parameters readily available. Suppose you only know the **end-points of any one of the diameters** of the sphere and have no information on the center point or radius. You still have a way to frame its equation.

Say the diameter $AB$ of the sphere has the endpoints $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$. Clearly, the center $C(h,k,l)$ is given by the mid-point of $AB$, which is given by:

Once we know the center, we can obtain the **radius** as the distance $CA$ or $CB$, through the **distance formula**:

We can then insert this center and radius in the **standard sphere equation**.

## Equation of a sphere from its center and any known point on its surface

Similar to the case above, if we know the **center** $(h,k,l)$ and any point $P(p_x,p_y,p_z)$