Midpoint Calculator

Last updated: 2026-05-19

What is a midpoint?

The midpoint of a line segment is the point that lies exactly halfway between its two endpoints, dividing the segment into two equal parts. On a coordinate plane, each coordinate of the midpoint is the average of the corresponding coordinates of the endpoints. Entering the coordinates of both endpoints applies the midpoint formula and returns the x and y coordinates of the midpoint.

The midpoint formula

Given two points P₁(x₁, y₁) and P₂(x₂, y₂), their midpoint M is:

$$M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)$$

Each coordinate of the midpoint is the arithmetic mean of the corresponding coordinates of the two endpoints.

Why it equals the average

The midpoint lies at equal distance from both endpoints along each axis. For the x-coordinate, "equal distance" means:

$$M_x - x_1 = x_2 - M_x$$

Solving for M_x:

$$2M_x = x_1 + x_2 \quad\Rightarrow\quad M_x = \frac{x_1 + x_2}{2}$$

The same balancing argument applies to y independently. No trigonometry is needed — the midpoint is defined by the averaging condition on each coordinate.

Worked example

Consider the midpoint between P₁(3, −4) and P₂(9, 6).

$$M_x = \frac{3 + 9}{2} = \frac{12}{2} = 6$$ $$M_y = \frac{-4 + 6}{2} = \frac{2}{2} = 1$$

The midpoint is (6, 1). A check confirms it: the distance from P₁ to M equals the distance from M to P₂ under the distance formula.

Midpoint vs. centroid

The midpoint formula applies to a line segment (two endpoints). The centroid is the geometric center of a polygon — for a triangle with vertices A, B, C it is:

$$G = \left(\frac{x_A + x_B + x_C}{3},\; \frac{y_A + y_B + y_C}{3}\right)$$

The midpoint is a special case: the centroid of a two-point set equals the midpoint. For three or more points, divide by the number of points rather than by 2.

Extending to three dimensions and beyond

In three dimensions, simply average the z-coordinates as well:

$$M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2},\; \frac{z_1 + z_2}{2}\right)$$

Example: Midpoint between (2, 5, −1) and (8, 3, 7):

$$M = \left(\frac{2+8}{2},\; \frac{5+3}{2},\; \frac{-1+7}{2}\right) = (5,\; 4,\; 3)$$

The "average each coordinate" rule extends to any number of dimensions — it's the foundation of centroid calculations in linear algebra and machine learning (e.g., k-means clustering).

Common applications

• Segment bisection in geometry: Finding the midpoint of a chord, diagonal, or side.
• Construction and surveying: Locating the center of a wall, beam, or plot boundary.
• Computer graphics: Midpoint subdivision for Bézier curves and line clipping algorithms.
• Navigation: Finding a meeting point equidistant from two locations on a map.
• Data analysis: The midrange statistic (midpoint of the min/max range) uses the same formula.

Frequently Asked Questions (FAQ)

What is the midpoint formula?

The midpoint formula finds the point exactly halfway between two points P₁(x₁, y₁) and P₂(x₂, y₂): M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2). Each coordinate of the midpoint is the arithmetic mean of the corresponding coordinates of the two endpoints.

Is the midpoint the same as the centroid?

Not in general. The midpoint applies to a line segment (two points), while the centroid is the geometric center of a polygon or solid. For a triangle, the centroid is the average of all three vertices: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). The midpoint formula is a special case: the centroid of a two-point set equals the midpoint.

How do I find the midpoint in 3D?

Extend the formula by averaging the z-coordinates as well: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2). For example, the midpoint between (1, 2, 3) and (5, 6, 7) is (3, 4, 5). The same "average each coordinate" principle applies in any number of dimensions.

Why is the midpoint just the average of the coordinates?

The midpoint lies at equal distance from both endpoints along each axis. Equal distance along the x-axis means x_M − x₁ = x₂ − x_M, which solves to x_M = (x₁ + x₂) / 2 — the arithmetic mean. The same reasoning applies to the y-axis. No trigonometry or distance formula is needed because the midpoint is defined by the balancing condition on each coordinate independently.