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Sphere Equation Calculator.

Evaluate x² + y² + z² − r² for a sphere centered at the origin.

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02 / ANSWER

Radius

Result

value: -6.00000000

Use Cases

Check if a point lies on a sphere

Use this calculator to verify whether a given point (x, y, z) lies exactly on the surface of a sphere with a specified radius. The residual tells you the deviation.

Example: For a sphere of radius 5, check if point (3, 4, 0) is on the surface: residual = 9+16+0-25 = 0.

Compare distances in 3D space

When working with 3D coordinates, quickly evaluate how far a point is from a sphere's surface, which can be useful in geometry or physics problems.

Example: Compare two points to see which is closer to the sphere's surface by comparing their residuals.

Frequently Asked Questions

What does the Sphere Equation Calculator do?: It evaluates the residual of the sphere equation (x^2 + y^2 + z^2 - r^2) for given x, y, z, and radius. The residual indicates how far the point is from lying exactly on the sphere's surface.
How is the residual calculated?: The calculator computes x^2 + y^2 + z^2 - r^2. If the result is zero, the point lies exactly on the sphere. A positive or negative value indicates the point is outside or inside the sphere, respectively.
What units should I use for x, y, z, and radius?: Use any consistent unit (e.g., meters, feet, inches) for all inputs. The residual will be in squared units of that same unit, so it's best for relative comparisons.

Tips & Common Mistakes

Tips

Ensure all inputs are in the same unit system to get meaningful results.
A residual of zero means the point is exactly on the sphere; positive means outside, negative means inside.
Use the residual to compare relative distances: a smaller absolute value indicates a point closer to the surface.
Double-check your radius value; a common error is entering the diameter instead.

Common Mistakes to Avoid

Forgetting to square the radius: the equation uses r^2, not r.
Using inconsistent units for coordinates and radius, leading to incorrect residuals.
Misinterpreting the sign: positive residual means outside, negative means inside, not the other way around.

Last updated: August 13, 2026

Use Cases

Check if a point lies on a sphere

Use this calculator to verify whether a given point (x, y, z) lies exactly on the surface of a sphere with a specified radius. The residual tells you the deviation.

Example: For a sphere of radius 5, check if point (3, 4, 0) is on the surface: residual = 9+16+0-25 = 0.

Compare distances in 3D space

When working with 3D coordinates, quickly evaluate how far a point is from a sphere's surface, which can be useful in geometry or physics problems.

Example: Compare two points to see which is closer to the sphere's surface by comparing their residuals.

Frequently Asked Questions

What does the Sphere Equation Calculator do?

It evaluates the residual of the sphere equation (x^2 + y^2 + z^2 - r^2) for given x, y, z, and radius. The residual indicates how far the point is from lying exactly on the sphere's surface.

How is the residual calculated?

The calculator computes x^2 + y^2 + z^2 - r^2. If the result is zero, the point lies exactly on the sphere. A positive or negative value indicates the point is outside or inside the sphere, respectively.

What units should I use for x, y, z, and radius?

Use any consistent unit (e.g., meters, feet, inches) for all inputs. The residual will be in squared units of that same unit, so it's best for relative comparisons.

Tips & Common Mistakes

Tips

Ensure all inputs are in the same unit system to get meaningful results.
A residual of zero means the point is exactly on the sphere; positive means outside, negative means inside.
Use the residual to compare relative distances: a smaller absolute value indicates a point closer to the surface.
Double-check your radius value; a common error is entering the diameter instead.

Common Mistakes to Avoid

Forgetting to square the radius: the equation uses r^2, not r.
Using inconsistent units for coordinates and radius, leading to incorrect residuals.
Misinterpreting the sign: positive residual means outside, negative means inside, not the other way around.

Use Cases

Check if a point lies on a sphere

Use this calculator to verify whether a given point (x, y, z) lies exactly on the surface of a sphere with a specified radius. The residual tells you the deviation.

Example: For a sphere of radius 5, check if point (3, 4, 0) is on the surface: residual = 9+16+0-25 = 0.

Compare distances in 3D space

When working with 3D coordinates, quickly evaluate how far a point is from a sphere's surface, which can be useful in geometry or physics problems.

Example: Compare two points to see which is closer to the sphere's surface by comparing their residuals.

Frequently Asked Questions

What does the Sphere Equation Calculator do?

It evaluates the residual of the sphere equation (x^2 + y^2 + z^2 - r^2) for given x, y, z, and radius. The residual indicates how far the point is from lying exactly on the sphere's surface.

How is the residual calculated?

What units should I use for x, y, z, and radius?

Use any consistent unit (e.g., meters, feet, inches) for all inputs. The residual will be in squared units of that same unit, so it's best for relative comparisons.

Tips & Common Mistakes

Tips

Ensure all inputs are in the same unit system to get meaningful results.
A residual of zero means the point is exactly on the sphere; positive means outside, negative means inside.
Use the residual to compare relative distances: a smaller absolute value indicates a point closer to the surface.
Double-check your radius value; a common error is entering the diameter instead.

Common Mistakes to Avoid

Forgetting to square the radius: the equation uses r^2, not r.
Using inconsistent units for coordinates and radius, leading to incorrect residuals.
Misinterpreting the sign: positive residual means outside, negative means inside, not the other way around.