Geometry
Distance Formula Calculator
Find the straight-line distance between two points in 2D or 3D. Type any coordinates and the answer, the right triangle behind it, the midpoint, the slope, and every step update live.
Plane — two coordinates
x₁
y₁
x₂
y₂
Distance
d =5
Exact whole-number distance
The distance is the hypotenuse of a right triangle.
Δx
3
Δy
4
Midpoint
(1.5, 2)
Slope
1.333333
d²
25
Try an example
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The basics
What is the distance formula?
The distance formula returns the length of the straight line between two points on a coordinate plane. It is one of the most-used results in coordinate geometry — and it is really just the Pythagorean theorem, rewritten in terms of coordinates rather than triangle sides.
d = √((x₂ − x₁)² + (y₂ − y₁)²)
The 2D formula
Take the difference of the x-coordinates and the difference of the y-coordinates, square both, add, and take the square root. The result is the length of the segment joining the points.
(0, 0) → (3, 4): √(9 + 16) = 5
a² + b² = c²
It's the Pythagorean theorem
The horizontal gap Δx and vertical gap Δy form the two legs of a right triangle. The distance is the hypotenuse, so it follows straight from a² + b² = c².
legs Δx, Δy → hypotenuse d
Method
How to calculate it by hand
Four steps take you from two points to a distance.
- 1
Find the horizontal and vertical change
Subtract the matching coordinates: Δx = x₂ − x₁ and Δy = y₂ − y₁. Sign doesn't matter — it disappears in the next step.
Δx = x₂ − x₁, Δy = y₂ − y₁ - 2
Square each difference
Squaring removes any negative sign and sets up the Pythagorean sum.
(Δx)² and (Δy)² - 3
Add the squares
Sum them to get d² — the squared distance, also called the radicand.
(Δx)² + (Δy)² - 4
Take the square root
The square root of that sum is the distance. Round only at the very end.
d = √((Δx)² + (Δy)²)
Worked examples
See it worked out
The same procedure handles a clean Pythagorean triple, negative coordinates, and a third dimension. Each example is computed live, not hardcoded.
Given P = (0, 0) and Q = (3, 4)
Substitute: d = √((3−0)² + (4−0)²)
Square & sum: d = √(9 + 16) = √25
d = 5
The classic Pythagorean triple — a whole-number distance of 5.
Where it shows up
Why distance matters
Measuring the gap between two coordinates is a building block across navigation, graphics, and data.
map (x, y)
Navigation & maps
Straight-line (“as the crow flies”) distance between two locations on a grid is the first estimate every routing system computes.
hit test
Games & graphics
Collision detection and range checks ask whether two objects are closer than some radius — a distance comparison every frame.
‖p − q‖
Data & clustering
Euclidean distance scores how similar two data points are; it powers k-means clustering and nearest-neighbor search.
Going further
From the plane to space
The formula generalizes cleanly. Add one squared term per dimension — the structure never changes. Switch the tool above to 3D to try it.
d = |x₂ − x₁|
1D — on a line
On a number line the distance is just the absolute difference of the two values.
d = √(Δx² + Δy²)
2D — in a plane
Two coordinates per point: the standard distance formula and a right triangle.
d = √(Δx² + Δy² + Δz²)
3D — in space
A third squared term for depth gives the distance between two points in 3D space.
Quick chart
Distance from the origin to common points
A fast lookup for the distance from (0, 0) to nearby points. The exact tag marks whole-number distances — the Pythagorean triples.
(0, 0) → (1, 0)
1
(0, 0) → (1, 1)
≈ 1.414
(0, 0) → (2, 1)
≈ 2.236
(0, 0) → (3, 4)
5
(0, 0) → (5, 12)
13
(0, 0) → (6, 8)
10
(0, 0) → (8, 15)
17
(0, 0) → (1, 2)
≈ 2.236
(0, 0) → (2, 2)
≈ 2.828
(0, 0) → (3, 3)
≈ 4.243
(0, 0) → (4, 3)
5
(0, 0) → (2, 3)
≈ 3.606
Worth knowing
Rules and edge cases
A handful of facts make the distance formula faster to use and harder to get wrong.
d(P, Q) = d(Q, P)
Order is irrelevant
Swapping the two points gives the same distance, because each difference is squared.
d ≥ 0
Never negative
Distance is always zero or positive. A result of zero means the two points coincide.
Δx = 0 → d = |Δy|
Same column or row
When the points share an x (or y) value, one leg vanishes and the distance is just the other leg.
round last
Round only at the end
Keep full precision through the squaring and the sum; round the single final value. Early rounding compounds error.
d² = Δx² + Δy²
Skip the root for comparisons
To find which of two points is closer, compare the squared distances. The root is monotonic, so the order is the same and you save a step.
mid = ((x₁+x₂)/2, …)
Midpoint averages, not subtracts
The midpoint averages the coordinates; the distance subtracts them. Don't mix the two formulas up.
Pitfalls
Common mistakes to avoid
Most wrong answers come from a few predictable slips — here is how to dodge them.
√(a + b) ≠ √a + √b
Don't split the square root
The root applies to the whole sum. You must add the squares first, then take a single square root over everything.
(−6)² = 36
Square the whole difference
A negative difference squares to a positive number. Forgetting the parentheses and writing −6² as −36 flips the sign and breaks the result.
x with x, y with y
Match the coordinates
Subtract x from x and y from y. Crossing them (x₂ − y₁) is a common and silent error.
d, not d²
Take the final root
Stopping at the sum of squares gives d², not the distance. The last square root is not optional.
FAQ
Common questions
The distance formula gives the straight-line distance between two points (x₁, y₁) and (x₂, y₂): d = √((x₂ − x₁)² + (y₂ − y₁)²). It measures the length of the segment joining them.
Subtract the x-coordinates to get the horizontal change, and the y-coordinates to get the vertical change. Square each difference, add them, then take the square root. For (0, 0) and (3, 4): √(3² + 4²) = √25 = 5.
It is the Pythagorean theorem in coordinates. The horizontal change (x₂ − x₁) and vertical change (y₂ − y₁) are the two legs of a right triangle, and the distance is the hypotenuse: d² = (x₂ − x₁)² + (y₂ − y₁)².
No. Each difference is squared, so (x₂ − x₁)² equals (x₁ − x₂)² and likewise for y. Swapping the two points gives exactly the same distance.
Add a third squared term for the z-axis: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). The method is identical — square every coordinate difference, add them, and take the square root. Switch the tool to 3D mode to compute it.
No. Distance is always zero or positive. Squaring each difference removes any sign, and the square root of a non-negative number is never negative. A distance of zero means the two points are the same.
The distance formula gives the length of the segment between two points. The midpoint formula gives the point exactly halfway along it: ((x₁ + x₂) / 2, (y₁ + y₂) / 2). The tool reports both at once.
Squaring does two jobs: it makes each leg positive regardless of direction, and it sets up the Pythagorean relationship d² = (Δx)² + (Δy)². The final square root then undoes the squaring to return an actual length.
Keep going
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The Distance Formula: How to Find the Distance Between Two Points
The distance between two points is d = √((x₂−x₁)² + (y₂−y₁)²). It comes straight from the Pythagorean theorem: the horizontal and vertical gaps between the points are the two legs of a right triangle, and the distance is the hypotenuse.
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Written by
UC
Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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