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Distance Formula Calculator

Distance between two points, with the exact radical.

Inputs

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Formula

d = √((x₂ − x₁)² + (y₂ − y₁)²)

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

The horizontal and vertical changes between the points are the two legs of a right triangle, and the distance is its hypotenuse — so this is the Pythagorean Theorem written in coordinates. When the coordinates are whole numbers the answer is also given as a simplified radical such as 2√5, which is what most textbooks expect.

Worked example

From (1, 2) to (7, 10) the change is 6 across and 8 up. 6² + 8² = 36 + 64 = 100, and √100 = 10 — a 6-8-10 right triangle.

Where this can give the wrong answer

  • The exact radical form appears only when Δx² + Δy² is a whole number. With decimal coordinates there is no tidy radical, so only the decimal is shown.
  • Squaring removes the sign, so it does not matter which point is first or whether a change is negative.
  • This is straight-line distance on a flat plane. For three dimensions add (z₂ − z₁)² under the root; for places on a map use a great-circle formula instead.

FAQ

They are the same number. 2√5 is the exact form — √20 with the perfect square 4 pulled out — and 4.4721 is its decimal rounded to four places. The page shows both when the coordinates are whole numbers.

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