Distance Formula Calculator

How to use the distance formula calculator

The distance formula is the Pythagorean theorem wearing coordinates. Draw the horizontal gap between two points and the vertical gap, and you have the two short sides of a right triangle whose hypotenuse is the distance you want. That is the whole idea, and it is why the formula never needs memorising separately from a² + b² = c².

Because both gaps are squared, their signs disappear before the root is taken. The two points can therefore be given in either order, and negative coordinates need no special treatment. From (−2, −3) to (2, 5) the gaps are 4 and 8 whichever end you start from, and the distance is the same either way.

Those three are worth comparing. A distance that lands on a whole number needs the two gaps to form a Pythagorean triple, and triples are scarce: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples. Every other pair of whole-number coordinates gives an irrational distance, so a radical is the normal case rather than the awkward one.

A third axis changes nothing structurally. In space the formula gains one more squared term and becomes the square root of Δx² + Δy² + Δz², which is the Pythagorean theorem applied twice — once in the base plane and once to lift that result to the point above it. The toggle above switches between the two.

Three different things are called the distance between two points

Search for the distance between two points and the results split three ways, because three unrelated quantities share the phrase. This page computes one of them, and knowing which is the difference between a right answer and a confidently wrong one.

The failure is quiet, which is what makes it worth naming. Feed a latitude and longitude into the formula here and it returns a number rather than an error — a meaningless one, in degrees, which is not a unit of length. Nothing about the output will tell you the wrong tool was used.

Why it is called Euclidean distance

The name distinguishes it from other ways of measuring how far apart two points are. Euclidean distance goes straight through the space between them. Manhattan distance, also called taxicab distance, goes along the axes instead, the way a taxi crosses a grid of streets rather than driving through the buildings.

The distinction matters beyond geometry. Machine learning uses both as measures of similarity between records, and they can rank the same set of neighbours differently. Euclidean is the default almost everywhere, which is precisely why the name is worth attaching to it rather than calling it just the distance.

The idea also extends past three axes without changing shape. For points described by many numbers rather than two or three, the Euclidean distance is still the square root of a sum of squared differences, with one term per number. This page handles two axes and three, which is where the geometry stays drawable.

One distance, and infinitely many points that give it

The formula runs one way without complaint and refuses to run backwards, which is worth understanding rather than treating as a limitation. Two points give exactly one distance. One point and a distance give infinitely many second points, and no amount of algebra narrows that down, because the information genuinely is not there.

The set of them has a familiar shape. Every point sitting a distance d from a fixed point (x₁, y₁) satisfies (x − x₁)² + (y − y₁)² = d², which is the equation of a circle of radius d centred on that point. The distance formula and the equation of a circle are the same statement, rearranged.

In three dimensions the same rearrangement gives (x − x₁)² + (y − y₁)² + (z − z₁)² = d², which is a sphere rather than a circle. Adding an axis adds a squared term and nothing else, exactly as it does in the distance formula itself, because the two equations were never really different.

When a decimal is the wrong answer

Most distances between whole-number coordinates are irrational, meaning no decimal ever finishes writing them. The square root of 208 is not 14.4222; it is a number whose digits continue without repeating, and 14.4222 is where the rounding stopped. Writing an equals sign between them is false, and this page uses "about" instead.

Not every radicand simplifies. The square root of 13 stays as it is because 13 is prime and has no square factor to remove, and the same is true of √2 and √145. In those cases the radical shown is simply the root itself, and it is still the exact answer rather than the decimal beneath it.

The formula, worked line by line

One formula, and a third axis only lengthens the sum underneath the root. Nothing else about it changes between two dimensions and three.

That judgement is worth making deliberately rather than by habit. A radical is exact and belongs in an algebra answer; a decimal is measurable and belongs on a drawing. Neither is more correct in general, and the two are not interchangeable in either direction.

distance = √((x₂ − x₁)² + (y₂ − y₁)²)
in three dimensions = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)
midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Manhattan distance = |x₂ − x₁| + |y₂ − y₁|
simplified radical: pull the largest square factor out from under the root

Why the formula cannot go wrong by a sign

Every term under the root is a square, and a square is never negative. That makes the sum non-negative, the root real, and the distance zero only when the two points coincide. There is no arrangement of inputs that produces a negative distance or an imaginary one, which is not true of most formulas with a root in them.

The midpoint, which behaves quite differently

One consequence is a useful check: the midpoint should sit the same distance from each endpoint. If those two halves come out unequal, the arithmetic went wrong, because the midpoint is the only point on the segment for which they must match.

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