The Midpoint Calculator finds the exact central point equidistant from two endpoints on a number line, in 2D Cartesian plane coordinates $(x, y)$ , or in 3D Euclidean space $(x, y, z)$ . Finding midpoints is fundamental to geometry, algebra, civil engineering, computer graphics, and navigation.

What is a Midpoint?

In geometry, a midpoint is the point on a line segment that divides it into two congruent segments of equal length. Because it lies exactly halfway between the two endpoints, its coordinates are the simple arithmetic mean (average) of the corresponding coordinates of the endpoints.

Midpoint Formulas

1. 1D Number Line Midpoint

To find the center point between two values $a$ and $b$ : $M = \frac{a + b}{2}$

2. 2D Coordinate Geometry Midpoint

For two endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ : $M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$

3. 3D Space Coordinates Midpoint

For two endpoints in three-dimensional space $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ : $M = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2}, \; \frac{z_1 + z_2}{2} \right)$

4. Segment Length (Euclidean Distance)

The total distance $d$ between two points is given by the Pythagorean distance theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step-by-Step Worked Examples

Example 1: Finding 2D Midpoint Between Two Points

Find the midpoint between point $A(2, -4)$ and point $B(8, 6)$ :

Calculate the X-Coordinate: $x_m = \frac{2 + 8}{2} = \frac{10}{2} = 5$
Calculate the Y-Coordinate: $y_m = \frac{-4 + 6}{2} = \frac{2}{2} = 1$
Result: The midpoint $M$ is $(5, 1)$ .

Example 2: Finding a Missing Endpoint Given the Midpoint

Suppose the midpoint $M$ of segment $AB$ is $(4, 3)$ , and endpoint $A$ is $(1, 1)$ . Find endpoint $B(x_2, y_2)$ :

$4 = \frac{1 + x_2}{2} \implies 8 = 1 + x_2 \implies x_2 = 7$ $3 = \frac{1 + y_2}{2} \implies 6 = 1 + y_2 \implies y_2 = 5$

Endpoint $B$ is located at $(7, 5)$ .

Frequently Asked Questions

Can a midpoint have negative coordinates?

Yes. If coordinates lie in quadrants II, III, or IV of the Cartesian coordinate system, the average can result in negative values or zero (e.g., the midpoint between $(-4, 2)$ and $(2, -6)$ is $(-1, -2)$ ).

What is the difference between a midpoint and an average?

A midpoint is specifically the geometric center between two spatial locations or coordinates, whereas an average (mean) can apply to any set of two or more numbers. Geometrically, the midpoint is simply the coordinate-wise average of two endpoints.

Does the midpoint formula work for curved surfaces?

No. The standard midpoint formula assumes a flat Euclidean plane. For geographic coordinates (latitude and longitude) on Earth's spherical surface, navigation calculations require Great-Circle distance and spherical trigonometry.

Midpoint Calculator