Engineering
Distance formula: 2D and 3D Euclidean vs Manhattan grid distance
Calculate straight-line Pythagorean distance in 2D or 3D space with SQRT, and compare right-angle warehouse aisle distance with ABS.
Straight-line Pythagorean vs right-angle aisle distance
Between points (X1, Y1) = (2, 3) and (X2, Y2) = (5, 7), horizontal delta is 3 and vertical delta is 4. Straight-line Euclidean distance equals SQRT((X2 - X1)^2 + (Y2 - Y1)^2) = SQRT(9 + 16) = 5.
Adding vertical rack height (Z2 - Z1)^2 inside SQRT extends the exact same formula to 3D coordinates (SQRT(3^2 + 4^2 + 12^2) = SQRT(169) = 13), while ABS(X2 - X1) + ABS(Y2 - Y1) computes right-angle Manhattan travel along rectangular aisles (3 + 4 = 7).
Use Cartesian coordinates in the same units, not latitude/longitude degrees. Manhattan distance assumes unobstructed axis-aligned travel; real obstacles and permitted routes can make actual travel longer.
| Point 1 (X1,Y1,Z1) | Point 2 (X2,Y2,Z2) | 2D Euclidean | 3D Euclidean | 2D Manhattan |
|---|---|---|---|---|
| (2, 3, 0) | (5, 7, 12) | 5 | 13 | 7 |
| (2, 3, 0) | (2, 3, 0) | 0 | 0 | 0 |
2D, 3D, and Manhattan distance formulas
Parenthesize each coordinate difference before squaring so the exponent applies to the whole difference. This also avoids relying on engine-specific unary-minus precedence.
=ROUND(SQRT((D2 - A2)^2 + (E2 - B2)^2), 4)3D Euclidean distance formula
Include (F2 - C2)^2 inside SQRT for 3D CAD, drone, or warehouse rack coordinates.
=ROUND(SQRT((D2 - A2)^2 + (E2 - B2)^2 + (F2 - C2)^2), 4)Try the example
Use the original values or change them to test your understanding.