Distance Formula Calculator
Enter two coordinate points to find their straight-line distance, exact radical, decimal value, coordinate changes, and midpoint with clear steps.
Exact: √25
- Horizontal change: 3 − 0 = 3.
- Vertical change: 4 − 0 = 4.
- d = √(3² + 4²) = 5.
All calculations run locally in your browser. Your entered values are not sent to an external service.
How to Use the Distance Formula Calculator
Enter the x-coordinate and y-coordinate for each point. Keep each coordinate pair together. Select Calculate Distance or press Enter. The calculator reports the straight-line distance, coordinate changes, squared distance, midpoint, exact radical, decimal value, and substitution steps.
- Label the points (x₁, y₁) and (x₂, y₂).
- Enter all four values, including any negative signs.
- Calculate the horizontal change x₂ − x₁ and vertical change y₂ − y₁.
- Square both changes, add them, and take the nonnegative square root.
Swapping the two points does not change distance. Both coordinate differences reverse sign, but squaring removes the signs. If both points are identical, the distance is zero, which is a valid result.
Distance Formula Between Two Points
The formula comes from the Pythagorean theorem. The horizontal coordinate difference forms one leg of a right triangle. The vertical difference forms the other leg. The segment joining the points is the hypotenuse.
Because both changes are squared, the value under the square root cannot be negative for real coordinates. Distance itself is always zero or positive. Direction is not part of distance, so moving from point A to point B has the same length as moving from B to A.
Exact radical and decimal answer
An exact answer may remain as a square root, such as √34. A decimal such as 5.83095189485 is an approximation. Keep the radical during algebra work and round the decimal only at the end according to the precision required by the problem.
Squared distance
Distance squared equals Δx² + Δy². Comparing squared distances can be useful because it avoids square roots. Since the square-root function increases with its input, the pair with the smaller squared distance also has the smaller actual distance.
Distance Formula Examples
A 3-4-5 triangle
Between (0, 0) and (3, 4), Δx = 3 and Δy = 4. Then d = √(3² + 4²) = √25 = 5. This familiar Pythagorean triple produces a whole-number distance.
Two general points
Between (2, 3) and (7, 6), Δx = 5 and Δy = 3. The distance is √(25 + 9) = √34, approximately 5.83095189485 units.
Negative coordinates
Between (−2, −3) and (4, 5), the horizontal change is 6 and the vertical change is 8. The distance is √(36 + 64) = √100 = 10. Negative coordinates require careful subtraction, but the formula does not change.
Decimal coordinates
Between (1.5, 2.5) and (4.5, 6.5), Δx = 3 and Δy = 4, so the distance is 5. Decimal coordinates can still produce an exact whole-number distance.
The same point twice
Between (7, 7) and (7, 7), both changes equal zero. The distance is √0 = 0. Unlike slope, which needs two distinct points to identify one line, distance can be calculated for identical points.
How to Interpret Coordinate Distance
The result is the shortest straight-line length between two points in a flat coordinate plane. It is not the travel distance along streets, paths, or curved surfaces. Units come from the coordinate axes, so coordinates measured in meters produce a distance in meters.
Coordinate distance is used in geometry, mapping, graphics, physics, data analysis, and engineering. It can measure the length of a line segment, compare nearby locations, calculate a circle radius, or check whether two points lie the same distance from a center.
Distance and midpoint
The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). It divides the segment into two equal parts. Each endpoint is half the full distance from the midpoint. The calculator reports it as a useful companion result, but midpoint and distance answer different questions.
Distance and slope
Slope measures direction and rate of change, while distance measures length. Two segments can have the same distance but different slopes. A vertical segment has a valid distance even though its slope is undefined. Use the slope calculator when direction is the main goal.
Three-dimensional extension
For points in three dimensions, add the squared z-coordinate change: d = √(Δx² + Δy² + Δz²). This page handles two-dimensional coordinates only. Geographic latitude and longitude also require a spherical or ellipsoidal model rather than this flat-plane formula over large areas.
Common Distance Formula Mistakes
Mixing coordinate pairs
Keep x₁ with y₁ and x₂ with y₂. Do not combine an x-value from one point with the y-value from the other.
Forgetting parentheses around negatives
Write negative coordinates in parentheses before subtracting. For example, 4 − (−2) equals 6.
Adding before squaring
Square the horizontal and vertical changes separately, then add the squares.
Reporting a negative distance
The principal square root is nonnegative. Distance cannot be negative.
Rounding too early
Keep the exact radical or full calculator value through intermediate work, then round once at the end.
Using planar distance for long geographic routes
Latitude and longitude on Earth require a geographic distance formula, especially across large regions.
Distance Formula Calculator FAQs
What is the distance formula?
It is d = √((x₂ − x₁)² + (y₂ − y₁)²) for two points in a coordinate plane.
Can distance be negative?
No. Coordinate distance is always zero or positive.
Can I use negative or decimal coordinates?
Yes. The calculator accepts finite positive, negative, zero, and decimal coordinates.
What happens when both points are the same?
Both coordinate changes are zero, so the distance is zero.
Does switching the points change the distance?
No. Reversing the points changes the signs of both differences, but their squares remain the same.
Is this the same as driving distance?
No. It is straight-line Euclidean distance in a flat coordinate plane.
Are my coordinates sent to a server?
No. The calculation runs locally in your browser.
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Browse Algebra ToolsReviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.