Distance Calculator — Free Online Distance Tool

How to Use the Distance Calculator

1. Select the dimension: Choose "2D Distance" for points on a flat plane (x, y coordinates) or "3D Distance" for points in three-dimensional space (x, y, z coordinates). The calculator dynamically shows z-coordinate input fields only when 3D mode is selected.
2. Enter Point 1 coordinates: Input the x₁ and y₁ values (and z₁ for 3D) for the first point. Values can be positive, negative, or zero. The coordinate fields are arranged side by side for quick entry.
3. Enter Point 2 coordinates: Input the x₂ and y₂ values (and z₂ for 3D) for the second point. The distance will be zero if both points are identical.
4. Review the results: The calculator instantly shows the distance between the points, the midpoint coordinates, a step-by-step formula showing all substitutions and intermediate calculations, and the individual coordinate differences (Δx, Δy, and Δz in 3D mode).

The default values (1, 2) and (7, 10) produce a distance of 10 units, which is a classic example demonstrating the distance formula. Modify any coordinate to see results update in real time.

Distance Formula

Variables Explained

• d (distance): The straight-line (Euclidean) distance between the two points. Always a non-negative value. Zero only when both points are identical.
• (x₁, y₁) and (x₂, y₂): The coordinates of the two points in 2D space. The subscripts 1 and 2 are arbitrary labels; the distance is the same regardless of which point you call "first."
• z₁, z₂: The z-coordinates (height or depth) of the two points in 3D space. Only relevant in 3D mode.
• Δx, Δy, Δz: The differences between corresponding coordinates. Δx = x₂ - x₁ represents the horizontal displacement, Δy = y₂ - y₁ the vertical displacement, and Δz = z₂ - z₁ the depth displacement.
• M (midpoint): The point that lies exactly halfway between the two given points, found by averaging each coordinate pair.

Step-by-Step Example

Find the distance between points (3, -2) and (9, 6):

1. Calculate Δx: 9 - 3 = 6
2. Calculate Δy: 6 - (-2) = 8
3. Square the differences: 6² = 36 and 8² = 64
4. Add the squares: 36 + 64 = 100
5. Take the square root: √100 = 10 units

The distance between (3, -2) and (9, 6) is exactly 10 units. The midpoint is ((3+9)/2, (-2+6)/2) = (6, 2). Notice that Δx = 6 and Δy = 8 form a 6-8-10 Pythagorean triple (a multiple of 3-4-5).

Practical Examples

Example 1: Marcus's Drone Flight Path

Marcus is operating a drone that needs to fly from a launch point at coordinates (10, 15, 0) meters to a target at (50, 75, 30) meters in a 3D space. He needs to calculate the direct flight distance to estimate battery consumption.

• Δx = 50 - 10 = 40 meters, Δy = 75 - 15 = 60 meters, Δz = 30 - 0 = 30 meters
• d = √(40² + 60² + 30²) = √(1,600 + 3,600 + 900) = √6,100 ≈ 78.10 meters
• At 5 meters/second cruise speed: 78.10 / 5 ≈ 15.6 seconds flight time

The direct flight distance is approximately 78.10 meters. Marcus's drone battery lasts for about 2 km of flight, so this short trip uses about 3.9% of his battery capacity. The 3D distance accounts for the 30-meter altitude gain, which would be missed by a 2D calculation.

Example 2: Sophie's Campus Walk

Sophie needs to walk from the library at position (2, 8) on her campus map (in hundreds of meters) to the science building at (7, 3). She wants to know the straight-line distance versus the grid-walking distance.

• Euclidean distance: √((7-2)² + (3-8)²) = √(25 + 25) = √50 ≈ 7.07 (hundreds of meters) = 707 meters
• Manhattan distance (walking on grid): |7-2| + |8-3| = 5 + 5 = 10 (hundreds of meters) = 1,000 meters
• Difference: 1,000 - 707 = 293 meters extra by walking the grid

If Sophie could walk in a straight line, the distance would be about 707 meters. Walking along the campus grid adds 293 meters (41% more). At a 5 km/h walking pace, the direct route takes about 8.5 minutes while the grid route takes about 12 minutes.

Example 3: Daniel's Pipe Installation

Daniel is a plumber running a pipe from a water heater in the basement at position (0, 0, -8) feet to a bathroom fixture at (12, 18, 2) feet in the building coordinate system. He needs the total pipe length for a diagonal run.

• Δx = 12, Δy = 18, Δz = 2 - (-8) = 10 feet
• d = √(12² + 18² + 10²) = √(144 + 324 + 100) = √568 ≈ 23.83 feet
• Adding 15% for fittings and bends: 23.83 × 1.15 ≈ 27.41 feet of pipe

Daniel needs approximately 27.41 feet of pipe to make the run with allowance for fittings. In practice, pipes cannot run perfectly diagonally through a building, so the actual route following walls and floors will be longer. The 3D calculation gives the absolute minimum distance.

Example 4: Olivia's Game Development

Olivia is developing a 2D game and needs to check if an enemy character at position (320, 180) pixels is within the player's attack range of 150 pixels. The player is at position (200, 100).

• d = √((320-200)² + (180-100)²) = √(120² + 80²) = √(14,400 + 6,400) = √20,800 ≈ 144.22 pixels
• 144.22 < 150, so the enemy IS within attack range
• Midpoint: ((200+320)/2, (100+180)/2) = (260, 140) for visual effects

The enemy is 144.22 pixels away, just within the 150-pixel attack range. Olivia can use the midpoint (260, 140) to spawn a visual effect between the player and enemy. This distance check using the Euclidean formula creates circular attack ranges, which feel more natural in games than square ranges based on coordinate differences.

Distance Reference Table

Tips and Complete Guide

Visualizing the Distance Formula

The best way to understand the distance formula is to visualize it as the Pythagorean theorem applied to a coordinate grid. Draw the two points on graph paper, then draw horizontal and vertical lines from each point to form a right triangle. The horizontal leg has length |x₂ - x₁| and the vertical leg has length |y₂ - y₁|. The distance you seek is the hypotenuse of this right triangle. This visual approach makes the formula intuitive and helps you verify calculations by estimating whether your answer seems reasonable given the point positions.

Applications in Computer Science

The distance formula is fundamental in computer science. Collision detection in video games checks distances between objects. Machine learning algorithms like k-nearest neighbors classify data points based on distance to training examples. Image processing uses distance calculations for edge detection and feature matching. Clustering algorithms group data points that are close together. Database systems use spatial indexes based on distance for efficient geographic queries. The computational cost is low, making it practical for real-time applications that check thousands of distance calculations per frame.

Distance on Curved Surfaces

The Euclidean distance formula assumes flat space. On curved surfaces like the Earth, the shortest path between two points is a great circle, and the distance must be calculated using the Haversine formula or Vincenty's formulae. For small distances (under a few kilometers), the flat-Earth approximation using the Euclidean formula is sufficiently accurate. For long distances, the curvature of the Earth becomes significant. At the equator, 1 degree of longitude is about 111 km, but this decreases toward the poles. Always consider whether curvature matters for your specific application.

Types of Distance Metrics

Besides Euclidean distance, several other distance metrics exist for different purposes. Manhattan distance (sum of absolute differences) models grid-based movement. Chebyshev distance (maximum absolute difference in any dimension) is used in chess for king movements. Minkowski distance generalizes both Euclidean and Manhattan distance with a parameter p. Cosine distance measures the angle between vectors rather than their spatial separation, useful in text analysis and recommendation systems. Choosing the right distance metric depends on your specific problem domain and what "closeness" means in your context.

Common Mistakes to Avoid

• Forgetting to square root: The formula gives d², not d. A common error is computing (x₂-x₁)² + (y₂-y₁)² and reporting that as the distance without taking the square root.
• Confusing 2D and 3D: When working in 3D, ensure you include the z-coordinate term. Omitting it underestimates the actual spatial distance. A point at (0,0,0) and (3,4,12) has a 3D distance of 13, not 5.
• Not handling negative coordinates: Negative coordinates are valid. The distance between (-5, 3) and (5, 3) is 10, not 0. Squaring eliminates negative signs, so the formula handles them correctly.
• Assuming distance equals displacement: Distance is a scalar (always positive), while displacement is a vector with direction. Two objects 10 units apart have a distance of 10 regardless of which direction you travel.
• Using wrong coordinates for midpoint: The midpoint formula averages corresponding coordinates. The midpoint of (2, 8) and (6, 4) is (4, 6), not (2, 4) or (6, 8). Average x with x and y with y.

Frequently Asked Questions