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 from its equation.
Are you wondering what the standard equation of a sphere is 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? Grab your favorite drink and keep reading; we'll tackle these questions together!
This calculator can also determine a sphere's surface area and volume from its equation. Want to calculate them without bothering with the sphere equation? Head to our sphere calculator!
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.
The sphere equation is a collection of all the points in 3-D space that lie equidistant from the center (h, k, l). When the center coincides with the origin, the equation simplifies to x² + y² + z² = r².
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). Inserting these into the equation above, we get:
Similarly, you can use the standard form to read off the radius and center of a sphere. For example, a sphere with the equation (x − 7)² + (y − 12)² + (z − 4)² = 36 has its center at (7, 12, 4), and its radius is r = √36 = 6.
❗ Be careful with the signs of the center coordinates. If the equation were instead (x − 7)² + (y + 12)² + (z + 4)² = 36, then the center would be (7, −12, −4).
How to derive the equation of a sphere
The sphere equation follows directly from the distance formula in three dimensions. A sphere is the set of all points (x, y, z) whose distance from the fixed center (h, k, l) equals the radius r:
Squaring both sides to remove the square root gives the familiar standard equation of a sphere:
Expanded form of the sphere equation
If you expand the squared terms and gather everything on one side, the standard equation turns into the expanded (general) form of a sphere:
where the new coefficients relate to the center and radius by:
- D = −2h, E = −2k, F = −2l; and
- G = h² + k² + l² − r².
To go the other way — from an expanded equation back to the center and radius — you complete the square for x, y, and z. That gives h = −D/2, k = −E/2, l = −F/2 and r² = h² + k² + l² − G. If r² turns out to be negative, the equation describes no real sphere; if it is exactly zero, it collapses to a single point (a "point sphere"). Our calculator handles all of these cases for you.
Equation of a sphere from the end-points of any diameter
Any diameter passes through the center, so if you know the two end-points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) of a diameter, the center is simply their midpoint:
The radius is half the distance between the two end-points, r = ½·√[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. Plug the center and radius into the standard form and you have the equation of the sphere.
Equation of a sphere from its center and any known point on its surface
If you know the center (h, k, l) and any point P(x, y, z) that lies on the surface of the sphere, then the radius is exactly the distance from the center to that point:
Once you have r, substitute the center and radius into the standard equation to complete the job.
How to use this equation of a sphere calculator
- Choose an input method from the drop-down: from center & radius, from center & a surface point, from the end-points of a diameter, or from an existing equation.
- Pick your unit system — Metric (meters) or US/Imperial (feet). The units label the radius, diameter, surface area, and volume; the equation itself is unit-free.
- Enter the values for the chosen method.
- Read the sphere equation in both standard and expanded form, together with the center, radius, diameter, surface area (4πr²), volume (⁴⁄₃πr³), and great-circle circumference (2πr).
Unit Systems
Metric System
Coordinates and the radius are treated in meters (m). Surface area is shown in m² and volume in m³.
US / Imperial System
Coordinates and the radius are treated in feet (ft). Surface area is shown in ft² and volume in ft³.
FAQ
What is the standard equation of a sphere?
It is (x − h)² + (y − k)² + (z − l)² = r², where (h, k, l) is the center and r is the radius.
How do I find the center and radius from a sphere's equation?
If the equation is already in standard form, the center is (h, k, l) and the radius is r = √(right-hand side). If it is in expanded form x² + y² + z² + Dx + Ey + Fz + G = 0, complete the square to get h = −D/2, k = −E/2, l = −F/2 and r = √(h² + k² + l² − G).
Can every equation of the form x² + y² + z² + Dx + Ey + Fz + G = 0 be a sphere?
No. Only when r² = h² + k² + l² − G is positive. If it equals zero the graph is a single point, and if it is negative there is no real sphere at all.
What is the surface area and volume of the sphere?
Once the radius r is known, the surface area is 4πr² and the volume is ⁴⁄₃πr³. This calculator reports both automatically.