GEOMETRYCoordinate GeometryMathematics Calculator
d

3D Distance Formula

The distance between two 3D points Pโ‚(xโ‚,yโ‚,zโ‚) and Pโ‚‚(xโ‚‚,yโ‚‚,zโ‚‚) is d = โˆš[(xโ‚‚โˆ’xโ‚)ยฒ+(yโ‚‚โˆ’yโ‚)ยฒ+(zโ‚‚โˆ’zโ‚)ยฒ]. Midpoint M = ((xโ‚+xโ‚‚)/2, (yโ‚+yโ‚‚)/2, (zโ‚+zโ‚‚)/2). Extends the 2D Pythagorean theorem.

Concept Fundamentals
d = โˆš(ฮ”xยฒ+ฮ”yยฒ+ฮ”zยฒ)
Distance
M = ((xโ‚+xโ‚‚)/2, (yโ‚+yโ‚‚)/2, (zโ‚+zโ‚‚)/2)
Midpoint
Generalizes 2D
3D
Straight-line length
Euclidean

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Pythagorean theorem in 3D. Midpoint = average of endpoints. Generalizes to n dimensions.

Key quantities
d = โˆš(ฮ”xยฒ+ฮ”yยฒ+ฮ”zยฒ)
Distance
Key relation
M = ((xโ‚+xโ‚‚)/2, (yโ‚+yโ‚‚)/2, (zโ‚+zโ‚‚)/2)
Midpoint
Key relation
Generalizes 2D
3D
Key relation
Straight-line length
Euclidean
Key relation

Ready to run the numbers?

Why: 3D distance is used in physics (particle separation), computer graphics (collision detection), robotics (path planning), and GIS (elevation-aware distance).

How: Apply d = โˆš[(xโ‚‚โˆ’xโ‚)ยฒ+(yโ‚‚โˆ’yโ‚)ยฒ+(zโ‚‚โˆ’zโ‚)ยฒ]. Midpoint: average each coordinate. Same as 2D formula with an extra z-term.

Pythagorean theorem in 3D.Midpoint = average of endpoints.
Sources:Khan Academy โ€” 3D GeometryWolfram MathWorld โ€” Distance

Run the calculator when you are ready.

Find 3D DistanceEnter two 3D points

Sample Examples

Input Coordinates

Point 1

Point 2

Results

3D Euclidean Distance
5.2
Straight-line distance between two 3D points
Distance
5.2
Midpoint
(2.5, 3.5, 4.5)

Visualization

Drag to rotate, scroll to zoom.

Calculation Steps

ฮ”x = 4 - 1 = 3, ฮ”y = 3, ฮ”z = 3

dยฒ = 3ยฒ + 3ยฒ + 3ยฒ = 27

d = โˆš27 = 5.2

Midpoint = ((1+4)/2, (2+5)/2, (3+6)/2) = (2.5, 3.5, 4.5)

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

d

d = โˆš[(xโ‚‚โˆ’xโ‚)ยฒ+(yโ‚‚โˆ’yโ‚)ยฒ+(zโ‚‚โˆ’zโ‚)ยฒ].

โ€” 3D Distance

M

Midpoint = average of coordinates.

โ€” Property

Key Takeaways

  • 3D distance extends the 2D Pythagorean theorem: d=sqrt(x2โˆ’x1)2+(y2โˆ’y1)2+(z2โˆ’z1)2d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}.
  • The midpoint in 3D is left(fracx1+x22,fracy1+y22,fracz1+z22right)\\left(\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2}, \\frac{z_1+z_2}{2}\\right).
  • Distance is always non-negative and zero only when points coincide.
  • This is Euclidean (straight-line) distance, not Manhattan or other metrics.
  • Used in physics, computer graphics, robotics, and navigation.

Did You Know?

GPS systems use 3D distance in Earth-centered coordinates.
Game engines compute millions of 3D distances per second for collision detection.
The formula generalizes to n dimensions for machine learning distance metrics.
Robotics uses 3D distance for path planning and obstacle avoidance.
Molecular dynamics simulations rely on interatomic distances.
VR/AR applications use 3D distance for spatial audio and interactions.

Understanding 3D Distance

The distance between two points in 3D space is the length of the straight line segment connecting them. It is the 3D extension of the Pythagorean theorem.

d=sqrt(x2โˆ’x1)2+(y2โˆ’y1)2+(z2โˆ’z1)2d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}
M=left(fracx1+x22,fracy1+y22,fracz1+z22right)M = \\left(\\frac{x_1+x_2}{2}, \\frac{y_1+y_2}{2}, \\frac{z_1+z_2}{2}\\right)

Expert Tips

For performance, compare squared distances when only relative order matters.
Normalize coordinates to avoid overflow with very large values.
Use the midpoint formula to find the center of a line segment in 3D.
In programming, sqrt is expensive; use dยฒ when possible for comparisons.

FAQ

Q: How does 3D distance relate to 2D?
A: 2D distance is the special case when zโ‚ = zโ‚‚ = 0.
Q: Can distance be negative?
A: No, distance is always โ‰ฅ 0.
Q: What is the midpoint formula in 3D?
A: Average each coordinate: M = ((xโ‚+xโ‚‚)/2, (yโ‚+yโ‚‚)/2, (zโ‚+zโ‚‚)/2).
Q: How is this used in games?
A: Collision detection, AI pathfinding, and LOD (level of detail) use 3D distance.
Q: What about curved space?
A: This formula assumes Euclidean (flat) space. Earth's surface uses great-circle distance.
Q: Can I use this for 4D or higher?
A: Yes, add more squared terms: d=sqrtsum(xi2โˆ’xi1)2d = \\sqrt{\\sum (x_{i2}-x_{i1})^2}.
Q: Why use squared distance sometimes?
A: Avoiding sqrt is faster; dยฒ preserves order for comparisons.

How to Use

  1. Enter coordinates (xโ‚, yโ‚, zโ‚) for point 1.
  2. Enter coordinates (xโ‚‚, yโ‚‚, zโ‚‚) for point 2.
  3. Click Calculate to get distance and midpoint.

Disclaimer

This calculator assumes Euclidean 3D space. For geographic coordinates, use appropriate geodesic formulas.

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