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3D Distance Calculator

Computes the Euclidean distance between two points in three-dimensional space, their midpoint, the coordinate differences, the direction cosines and direction angles of the connecting segment, and the Manhattan distance.

When to use

You have two (x, y, z) coordinates and need the straight-line distance, the halfway point or the direction of the line between them.

Do not use when: The points are in a plane (use distance-2d), you need vector products or the angle between two vectors (use vector-operations), or the points are geographic coordinates (use haversine-distance).

Formula

distance = √(Δx² + Δy² + Δz²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2); cos α = Δx / distance, cos β = Δy / distance, cos γ = Δz / distance; manhattan = |Δx| + |Δy| + |Δz|

Inputs

Parameter	Type	Required	Description
`x1`	number	yes	x-coordinate of the first point.
`y1`	number	yes	y-coordinate of the first point.
`z1`	number	yes	z-coordinate of the first point.
`x2`	number	yes	x-coordinate of the second point.
`y2`	number	yes	y-coordinate of the second point.
`z2`	number	yes	z-coordinate of the second point.

Outputs

Output	Type	Unit	Description
`distance`	number		√((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).
`midpoint_x`	number		(x₁ + x₂) / 2.
`midpoint_y`	number		(y₁ + y₂) / 2.
`midpoint_z`	number		(z₁ + z₂) / 2.
`midpoint`	string		The midpoint as a coordinate triple, e.g. '(2.5, 4, 9)'.
`delta_x`	number		x₂ − x₁.
`delta_y`	number		y₂ − y₁.
`delta_z`	number		z₂ − z₁.
`direction_cosines`	number_list		[Δx, Δy, Δz] / distance = [cos α, cos β, cos γ]; their squares sum to 1.
`direction_angles_degrees`	number_list	°	[α, β, γ]: angles between the segment (from point 1 to point 2) and the positive x, y and z axes.
`manhattan_distance`	number		\|Δx\| + \|Δy\| + \|Δz\|.

Example

(1, 2, 3) to (4, 6, 15): {"x1":1,"y1":2,"z1":3,"x2":4,"y2":6,"z2":15} → {"distance":13,"midpoint_x":2.5,"midpoint_y":4,"midpoint_z":9,"midpoint":"(2.5, 4, 9)","delta_x":3,"delta_y":4,"delta_z":12,"direction_cosines":[0.230769,0.307692,0.923077],"direction_angles_degrees":[76.6576,72.0798,22.6199],"manhattan_distance":19}

Origin to (2, 3, 6): {"x1":0,"y1":0,"z1":0,"x2":2,"y2":3,"z2":6} → {"distance":7,"midpoint":"(1, 1.5, 3)","direction_cosines":[0.285714,0.428571,0.857143],"manhattan_distance":11}

GET https://tttkmbb.com/api/v1/calculate/distance-3d?x1=1&y1=2&z1=3&x2=4&y2=6&z2=15

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/distance-3d (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/distance-3d · Markdown: https://tttkmbb.com/math/distance-3d.md · JSON definition: https://tttkmbb.com/math/distance-3d.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="distance-3d"
OpenAPI operationId: calculate_distance_3d
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Euclidean distance (reference)
Wikipedia – Direction cosine (reference)
Wolfram MathWorld – Direction Cosine (reference)

FAQ

What are direction cosines used for?

They are the components of the unit vector from point 1 to point 2, so they describe the orientation of the segment; cos²α + cos²β + cos²γ = 1 always holds.

Does the order of the points matter?

Not for distance or midpoint. Δx, Δy, Δz, the direction cosines and angles are taken from point 1 to point 2 and flip sign (angles become 180° − angle) if the points are swapped.

Related calculators

Distance Between Two Points Calculator — The planar version with Manhattan distance and midpoint.
Vector Calculator — Treat the two points as vectors: dot and cross products, angle.
Pythagorean Theorem Calculator — The distance formula is the Pythagorean theorem applied twice.