Distance between two points — coordinates in any length unit.
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
The distance between two points is derived directly from the Pythagorean theorem. If you draw a right triangle connecting point A and point B, the horizontal leg has length |x₂ − x₁| and the vertical leg has length |y₂ − y₁|. The hypotenuse — the straight-line distance — is found by squaring both legs, summing them, and taking the square root.
This is known as the Euclidean distance formula and works for any two points in a 2D Cartesian coordinate system.
d = √[(7 − 1)² + (10 − 2)²] d = √[(6)² + (8)²] d = √[36 + 64] d = √100 d = 10
Result: The distance between (1, 2) and (7, 10) is **10 units**.
The result 10 units means the shortest straight-line path from point (1, 2) to point (7, 10) is exactly 10 units long. Notice that this is a classic 6-8-10 Pythagorean triple (a scaled version of the 3-4-5 right triangle), which is why the answer comes out to a whole number. In most real-world problems the result will be an irrational number — the calculator returns the decimal approximation rounded to a reasonable number of significant figures.
The distance formula is one of the most fundamental tools in coordinate geometry. It is a direct application of the Pythagorean theorem (a² + b² = c²), where the two legs of the right triangle are the horizontal and vertical separations between your two points, and the hypotenuse is the distance you're solving for.
The relationship was established by the ancient Greek mathematician Euclid and later formalized in Cartesian coordinates by René Descartes in the 17th century. The coordinate plane itself is sometimes called the Cartesian plane in his honor.
The formula extends naturally to three dimensions:
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Add a third squared difference under the radical for the z-axis.
The midpoint M between (x₁, y₁) and (x₂, y₂) is M = ((x₁+x₂)/2, (y₁+y₂)/2). It's the exact center of the line segment connecting the two points.
The perpendicular distance from point (x₀, y₀) to the line ax + by + c = 0 is given by d = |ax₀ + by₀ + c| / √(a² + b²). This is different from the point-to-point distance formula.
They are the same relationship expressed differently. If you plot two points, draw a horizontal line from one and a vertical line from the other to form a right triangle, the hypotenuse of that triangle is the distance between the points — which is exactly what a² + b² = c² calculates.
Manhattan distance (also called taxicab distance) is |x₂−x₁| + |y₂−y₁| — the sum of the absolute differences, not the straight-line diagonal. It models travel on a grid (like city blocks) where diagonal movement is not allowed.
No. Because both differences are squared in the formula, the result is the same regardless of which point you enter first. d(A,B) always equals d(B,A).
Yes. The formula works perfectly with negative x or y values. Squaring the differences eliminates any negative signs, so (−3 − 5)² = (−8)² = 64, the same as (5 − (−3))² = 64.
The calculator returns a result in whatever units your input coordinates represent. If your coordinates are in meters, the distance is in meters. If they're in pixels, the distance is in pixels. The formula itself is unit-agnostic.
Yes. The straight-line distance between two points in a 2D plane is the Euclidean distance. The term 'Euclidean' distinguishes it from other distance metrics like Manhattan distance or Chebyshev distance.
Not directly. GPS coordinates are latitude/longitude values on a sphere, and using the flat-plane distance formula on them introduces error. For geographic distances, you would need the Haversine formula or a dedicated map distance tool.
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