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3D Distance Calculator

Finding the distance between two points in 3D space is fundamental in geometry, physics, engineering, and computer graphics. Whether you're measuring spatial separation in a CAD model, calculating distances in game development, or solving physics problems, the 3D distance formula extends the Pythagorean theorem into three dimensions. Enter the coordinates of both points to instantly determine the straight-line distance between them.

Last updated: May 10, 2026

Creators Anna Szczepanek, PhD Mathematics

Reviewers Mateusz Mucha and Jasmine J. Mah

8,650 people find this calculator helpful

The 3D Distance Formula

Measuring the distance between two points in three-dimensional space requires extending the familiar 2D distance concept into the z-axis. The formula calculates the straight-line (Euclidean) distance by accounting for differences in all three coordinate directions.

d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

d — The straight-line distance between the two points
x₁, y₁, z₁ — Coordinates of the first point in 3D space
x₂, y₂, z₂ — Coordinates of the second point in 3D space

How to Calculate 3D Distance

The calculation process involves six straightforward steps:

Identify the coordinates of your first point as (x₁, y₁, z₁)
Identify the coordinates of your second point as (x₂, y₂, z₂)
Calculate the difference in each dimension: Δx = x₂ − x₁, Δy = y₂ − y₁, Δz = z₂ − z₁
Square each difference: Δx², Δy², Δz²
Sum the three squared values
Take the square root of the sum to obtain the distance

Example: For points (1, 1, 1) and (3, 6, 9):

Δx = 3 − 1 = 2
Δy = 6 − 1 = 5
Δz = 9 − 1 = 8
d = √(4 + 25 + 64) = √93 ≈ 9.644

Understanding 3D Coordinates

Three-dimensional coordinates represent positions in space using three perpendicular axes: the x-axis (horizontal left-right), the y-axis (vertical up-down), and the z-axis (depth forward-backward). This system, called Cartesian coordinates, allows you to pinpoint any location in 3D space uniquely.

The origin, located at (0, 0, 0), serves as the reference point. For instance, the distance from any point (x, y, z) to the origin is simply √(x² + y² + z²). A point at (1, 1, 1) sits at distance √3 ≈ 1.732 units from the origin.

Common Pitfalls and Practical Considerations

Avoid these frequent mistakes when computing 3D distances:

Mixing up coordinate order — Ensure you're consistent about which point is point 1 and which is point 2. Since the formula squares the differences, order doesn't affect the final result, but confusion can lead to arithmetic errors during intermediate steps.
Forgetting to square differences — The squares in the formula are essential—they ensure all contributions are positive and give greater weight to larger separations. Omitting this step or squaring only some differences will produce an incorrect result.
Neglecting units and scale — If your coordinates use different units (metres vs. centimetres, for example), convert them to a common unit first. The calculated distance will match whatever unit system your coordinates use.
Assuming 2D when data is 3D — When coordinates include a z-value, you must use the 3D formula, not the 2D variant. Using the wrong formula is a common oversight that produces significantly incorrect distances.

Real-World Applications

3D distance calculations appear across diverse fields:

Computer graphics and gaming: Determining positions of objects, camera placement, collision detection, and visibility culling
Robotics and navigation: Calculating path lengths, obstacle avoidance, and spatial positioning
Physics simulations: Computing gravitational forces, electrostatic interactions, and particle movements
CAD and engineering: Verifying component spacing, assembly clearances, and tolerance requirements
Astronomy: Measuring distances between celestial objects within coordinate systems

Frequently Asked Questions

How does the 3D distance formula relate to the Pythagorean theorem?

The 3D distance formula is an extension of the Pythagorean theorem into three dimensions. In 2D, you'd use d = √[(x₂−x₁)² + (y₂−y₁)²]. The 3D version simply adds a third term for the z-axis difference. You can think of it as applying the Pythagorean theorem twice: first to combine the x and y differences, then again to combine that result with the z difference.

Can I calculate the distance if my points have negative coordinates?

Yes, absolutely. Negative coordinates work perfectly with this formula because you're squaring the differences. Whether a difference is positive or negative, squaring it always yields a positive result. For example, the distance from (−2, 3, 1) to (1, −1, 4) involves differences of 3, −4, and 3 respectively, which square to 9, 16, and 9, giving a distance of √34 ≈ 5.831 units.

What's the difference between 3D Euclidean distance and other distance metrics?

Euclidean distance is the straight-line distance you'd measure with a ruler. Other metrics, like Manhattan distance (sum of absolute differences) or Chebyshev distance (maximum absolute difference), measure differently and produce different results. For most spatial applications in physics, graphics, and engineering, Euclidean distance is the standard because it reflects actual geometric separation in continuous space.

Do I need to worry about floating-point precision errors?

For most practical purposes, no. Standard double-precision arithmetic handles 3D distance calculations accurately. However, if you're working with very large coordinate values (beyond 10¹⁵) or performing millions of calculations, accumulated rounding errors might matter. In such cases, consider using higher-precision arithmetic libraries or restructuring your calculations to minimize rounding.

How do I find the distance from a point to the origin?

Since the origin is located at (0, 0, 0), simply set x₁, y₁, and z₁ to zero. The formula simplifies to d = √(x₂² + y₂² + z₂²). For the point (3, 4, 0), the distance to the origin is √(9 + 16 + 0) = 5 units. This is useful in physics and engineering when measuring how far something sits from a reference point.

What if my coordinates are given in spherical or cylindrical form?

You'd need to convert them to Cartesian (x, y, z) coordinates first. Spherical coordinates (r, θ, φ) convert using x = r·sin(φ)·cos(θ), y = r·sin(φ)·sin(θ), z = r·cos(φ). Cylindrical coordinates (ρ, θ, z) convert using x = ρ·cos(θ) and y = ρ·sin(θ). Once you have Cartesian coordinates, use this calculator to find the distance between your two points.

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