Distance Between Points Calculator

Math Geometry • Coordinate Geometry

Written by STEM Calculators Team Published January 23, 2026 Updated February 24, 2026

Compute the Euclidean distance between two points in 2D or 3D: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Enter two points and click Calculate.

Point diagram (pan/zoom enabled)

Tip: in 3D mode, rotate the view to clearly see separation along the \(z\) direction.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Distance Between Two Points (2D and 3D)

The Euclidean distance is the straight-line distance between two points. It comes directly from the Pythagorean theorem: in 2D you form a right triangle with legs \(\Delta x\) and \(\Delta y\); in 3D you extend the idea by adding \(\Delta z\).

2D distance formula

For points \(A(x_1,y_1)\) and \(B(x_2,y_2)\):

\[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]

Define the coordinate differences: \(\Delta x=x_2-x_1\), \(\Delta y=y_2-y_1\). Then \(d=\sqrt{(\Delta x)^2+(\Delta y)^2}\).

3D distance formula

For points \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\):

\[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \]

Here \(\Delta z=z_2-z_1\), so the distance is \(d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}\).

Worked example (3D)

Let \(A(1,2,3)\), \(B(4,6,8)\). Then \(\Delta x=3\), \(\Delta y=4\), \(\Delta z=5\), and:

\[ d=\sqrt{3^2+4^2+5^2}=\sqrt{9+16+25}=\sqrt{50}\approx 7.07 \]

Common uses

Coordinate geometry: segment lengths, triangle side checks, perimeters.
3D geometry: distances in space, diagonals, point-to-point separations.
Physics: displacement magnitude between positions (when motion is a straight line).
Maps/GPS: approximate straight-line distance (not accounting for Earth curvature).

Notes and pitfalls

The Euclidean distance is always nonnegative: \(d\ge 0\). It equals \(0\) only when the two points are identical.
Units: if coordinates are in meters, then distance is in meters; if coordinates are in kilometers, distance is in kilometers.
For very large coordinates, rounding may occur—this is normal in floating-point arithmetic.

Quick reference

\[ \Delta x=x_2-x_1,\quad \Delta y=y_2-y_1,\quad \Delta z=z_2-z_1 \] \[ d_{2D}=\sqrt{(\Delta x)^2+(\Delta y)^2},\qquad d_{3D}=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2} \]

Frequently Asked Questions

What is the distance formula between two points in 2D?

For A(x1, y1) and B(x2, y2), the Euclidean distance is d = sqrt((x2 - x1)^2 + (y2 - y1)^2). It comes from the Pythagorean theorem using the horizontal and vertical differences.

How do you calculate distance between two points in 3D?

For A(x1, y1, z1) and B(x2, y2, z2), compute dx = x2 - x1, dy = y2 - y1, dz = z2 - z1, then d = sqrt(dx^2 + dy^2 + dz^2).

Can I use pi or scientific notation in the inputs?

Yes. Inputs accept entries like pi, sqrt(2), and 1e-3, and the calculator evaluates them before applying the distance formula.

Why is the distance always nonnegative?

The formula squares the coordinate differences and then takes a square root, so the result cannot be negative. The distance is 0 only when the two points are identical.

What units is the distance reported in?

The distance uses the same units as your coordinates. If x, y, and z are measured in meters, the computed distance is in meters as well.

Rate this calculator

Frequently Asked Questions

Related calculators