Distance formula calculator

The distance between points (x₁, y₁) and (x₂, y₂) comes from the Pythagorean theorem:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Example: A(1, 2) and B(4, 6).

• Δx = 4 − 1 = 3, Δy = 6 − 2 = 4
• d = √(9 + 16) = √25 = 5

The horizontal and vertical changes are the legs of a right triangle, and the distance is its hypotenuse. The calculator shows this working along with every result. It runs entirely in your browser.

The midpoint is the average of the coordinates:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

For A(1, 2) and B(4, 6): M = (2.5, 4).

Midpoints are handy for finding the center of a segment, the center of a circle from the ends of its diameter, or a fair meeting point on a grid map.

Slope: m = (y₂ − y₁) / (x₂ − x₁), the rise over the run. For our points, m = 4 / 3 ≈ 1.3333.

Slope-intercept form: y = mx + b, where b = y₁ − m × x₁. Here b = 2 − 4/3 ≈ 0.6667, so y = 1.3333x + 0.6667.

Angle from the x-axis: arctan(m) ≈ 53.13°.

Special cases: if the two y values are equal, the line is horizontal (slope 0, y = constant). If the x values are equal, the line is vertical, the slope is undefined and the equation is x = constant.

• Geometry homework: side lengths, perimeters and checking whether a shape is a parallelogram or a right triangle.
• Maps and games: straight-line distance between grid positions.
• Graphic design and CAD: distances between points in pixel or drawing coordinates.
• Physics: displacement between two positions.

On a city street grid, the walking distance is |Δx| + |Δy| (here 7 blocks) rather than the straight-line 5, because you cannot cut through buildings.

• Order does not matter for distance: squaring removes the sign. For slope, keep the same order in the numerator and denominator.
• Watch negative coordinates. 2 − (−3) = 5, not −1.
• Units: the distance is in the same unit as the coordinates. On real maps, latitude and longitude are not a flat grid, so this formula is only an approximation over short distances.
• Exactly equal points give a distance of 0 and no line.

To solve the triangle formed by three points, use the triangle calculator.

In 3D space, add the third coordinate: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). For example, the diagonal of a box 3 × 4 × 12 is √(9 + 16 + 144) = 13. This calculator works in two dimensions; for 3D you can apply the formula with the square root calculator.

The midpoint idea also extends: average each coordinate separately.

Checking a right angle: three points form a right triangle if the squared distances satisfy a² + b² = c². Find the three distances here, then compare. Two lines are perpendicular when the product of their slopes is −1, as with slopes 2 and −1/2. Parallel lines have equal slopes, so you can also check whether opposite sides of a four-sided shape are parallel. All of this runs in your browser as you type.