Distance and Midpoint Calculator

How it works

The distance between two points is the hypotenuse of a right triangle whose legs are the difference in x and the difference in y (and, in 3D, the difference in z): the Pythagorean theorem gives √(Δx² + Δy² + Δz²). The midpoint is simply the average of the two points, coordinate by coordinate.

The slope of the segment is rise over run, Δy ÷ Δx, and the angle is the direction of the segment measured counter-clockwise from the positive x axis. A vertical segment has no slope (division by zero) and an angle of 90°; both are defined only in two dimensions.

The section formula generalises the midpoint: the point that divides the segment in the ratio m : n, counting m parts from the first point and n parts from the second, is the weighted average (n·P₁ + m·P₂) ÷ (m + n). With m = n it is the midpoint; with m = 1, n = 2 it is one third of the way from the first point.

Formula

distance = √((x₂ − x₁)² + (y₂ − y₁)² [+ (z₂ − z₁)²])
midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2 [, (z₁ + z₂) ÷ 2])
slope    = (y₂ − y₁) ÷ (x₂ − x₁)          angle = atan2(y₂ − y₁, x₂ − x₁)
m:n point = ((n·x₁ + m·x₂) ÷ (m + n), (n·y₁ + m·y₂) ÷ (m + n) [, …])

Example

From (1, 2) to (4, 6): Δx = 3 and Δy = 4, so the distance is √(9 + 16) = 5 ft. The midpoint is ((1 + 4) ÷ 2, (2 + 6) ÷ 2) = (2.5, 4). The slope is 4 ÷ 3 = 1.3333 and the angle is atan(4 ÷ 3) = 53.13°. With the ratio 1 : 2 the dividing point is ((2 × 1 + 1 × 4) ÷ 3, (2 × 2 + 1 × 6) ÷ 3) = (2, 3.3333).

In 3D, from (1, 2, 3) to (4, 6, 15): Δ = (3, 4, 12), so the distance is √(9 + 16 + 144) = √169 = 13 and the midpoint is (2.5, 4, 9).

Assumptions and limitations

• Coordinates are Cartesian, on a flat plane or in ordinary space. For latitude and longitude use a great-circle distance instead.
• All coordinates share one unit, the one you choose, and the distance is in that unit. The unit is a label; nothing is converted.
• The m : n point is an internal division: both parts are positive and the point lies between the two ends.
• The angle is measured counter-clockwise from the positive x axis, between −180° and 180°.

Frequently asked questions