Midpoint Calculator - Average of Coordinates

The Midpoint Calculator averages x coordinates and y coordinates separately to find the center point of a segment. Enter two endpoints, or enter one endpoint with the midpoint to recover the other endpoint by reversing the average. The page redraws the segment with the midpoint marked.

The reference endpoints (2, 4) and (6, 8) give midpoint (4, 6). Related pages cover the Distance Calculator for segment length and the Slope Calculator for rise over run from the same two points. Those three tools share a two-point foundation but own different outputs.

Calculate the midpoint between two points

Midpoint is the point halfway along the segment in each coordinate direction. Each coordinate of the midpoint is the arithmetic mean of the matching coordinates at the ends. The Midpoint Calculator places that result on the line through the endpoints for any distinct pair.

M = ((x1 + x2) / 2, (y1 + y2) / 2)

Three-dimensional midpoints average z as well when z coordinates are present:

M = ((x1 + x2) / 2, (y1 + y2) / 2, (z1 + z2) / 2)

The midpoint divides the segment into two congruent halves of equal length. It is not the same as a point that splits the path at a different ratio; section formula with ratio m:n generalizes the average when the split is uneven.

Units follow the axes. If coordinates are in meters, the midpoint coordinates are in meters. Mixing units across axes breaks geometric meaning the same way mixed units break distance.

Apply the midpoint formula

Add corresponding coordinates and divide by two. This is equivalent to the average of the endpoints. The Midpoint Calculator expands the formula with the entered numbers and shows each coordinate step.

Mx = (x1 + x2) / 2
My = (y1 + y2) / 2

For (2, 4) and (6, 8):

Mx = (2 + 6) / 2 = 4
My = (4 + 8) / 2 = 6
midpoint = (4, 6)

Order of endpoints does not matter: averaging (6, 8) with (2, 4) still yields (4, 6). Negative coordinates work the same way. For (-2, 5) and (4, -1):

Mx = (-2 + 4) / 2 = 1
My = (5 + (-1)) / 2 = 2
midpoint = (1, 2)

On a number line (1D) the midpoint of a and b is (a + b) / 2, the same idea with a single coordinate.

Find an endpoint from the midpoint

If one endpoint and the midpoint are known, double the midpoint coordinate and subtract the known endpoint coordinate to find the missing end. This reverses the average. The Midpoint Calculator supports that recovery mode when worksheets give center and one tip of a segment.

x2 = 2 * Mx - x1
y2 = 2 * My - y1

If (2, 4) is one end and midpoint is (4, 6), the other end is:

x2 = 2 * 4 - 2 = 6
y2 = 2 * 6 - 4 = 8
endpoint = (6, 8)

The same reverse step works in 3D with z2 = 2 * Mz - z1. Geometrically, the known endpoint and the missing endpoint are symmetric with respect to the midpoint: the midpoint is the center of that point reflection.

A common error is to subtract the midpoint from the known end without doubling, which yields a vector to the midpoint rather than the far endpoint. Another is to average the midpoint with the known end, which pulls toward the center instead of reflecting through it.

Solve for points (2,4) and (6,8)

Apply the midpoint formula to the reference endpoints. Then verify the reverse step and the companion distance and slope for context. 1. Midpoint

M = (4, 6)

2. Recover endpoint from M and (2, 4)

(2 * 4 - 2, 2 * 6 - 4) = (6, 8)

3. Distance of the segment (cross-check)

d = sqrt((6-2)^2 + (8-4)^2) = sqrt(16 + 16) = sqrt(32) = 4 * sqrt(2) approx 5.66

4. Slope of the segment (cross-check)

m = (8 - 4) / (6 - 2) = 4 / 4 = 1

Distance and slope belong in detail on their own pages. Here they only confirm that (4, 6) sits halfway: from (2, 4) to (4, 6) the change is (2, 2), and from (4, 6) to (6, 8) the change is again (2, 2).

Section formula with ratio 1:1 is exactly the midpoint. For an uneven split m:n measured from the first point toward the second, the division point is ((n*x1 + m*x2)/(m+n), (n*y1 + m*y2)/(m+n)). When m = n the expression collapses to the average used here. Keep uneven ratios on worksheets that ask for them; this page owns the halfway case and the endpoint reflection.

Coordinate geometry proofs often show that the diagonals of a parallelogram bisect each other by verifying that both diagonals share the same midpoint. Entering the four vertices as two diagonal pairs should return one common midpoint when the quadrilateral is a parallelogram.

Use midpoint on a grid worksheet

On graph paper, the Midpoint Calculator matches counting boxes: from (2, 4) walk halfway toward (6, 8) by adding half of (4, 4), which is (2, 2), landing on (4, 6). The same half-step works with negative coordinates.

Plotting both endpoints and the midpoint should show equal segment lengths on each side when a ruler check is available.

For a missing endpoint problem, mark the known end and the midpoint first, then reflect: the vector from the known end to the midpoint equals the vector from the midpoint to the unknown end. Doubling the midpoint and subtracting the known end encodes that reflection in algebra. Checking that both half-segments have the same length with the Distance Calculator catches arithmetic slips.

Unit note: coordinates inherit the axis units. If x is in metres and y is in metres, the midpoint is in metres. Do not average a point given in centimetres with one given in metres without converting first.

Frequently asked questions

Summary

Midpoint Calculator returns the center of a segment from two endpoints and can recover a missing endpoint from one end and the midpoint, with substituted arithmetic and a labelled figure. Worked numbers match (2, 4) and (6, 8): midpoint (4, 6). Use the Distance Calculator and Slope Calculator for length and gradient from the same two-point inputs.