Midpoint Calculator

The midpoint of a segment is the point exactly halfway between its endpoints — the average of the coordinates.

2D, for $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$:

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

3D simply averages a third coordinate too:

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

The formula is symmetric: swapping the two points changes nothing, because addition is commutative. It assumes a straight segment in a Cartesian coordinate system, and it works for any real coordinates — negatives, fractions, decimals — with no special cases.

A useful check: the midpoint must lie between the endpoints in each coordinate. If your $x$-value is outside $[\min(x_1,x_2), \max(x_1,x_2)]$, you have made a sign error.

Finding a missing endpoint

Given one endpoint $A$ and the midpoint $M$, solve the midpoint equation for the other endpoint $B$:

$\frac{x_A + x_B}{2} = x_M \;\Longrightarrow\; x_B = 2x_M - x_A$

So the compact rule is

$B = 2M - A = (2x_M - x_A,\; 2y_M - y_A)$

Double, then subtract — not "add the difference once". The most common error here is computing $M - A$ and stopping.

Midpoint and distance together

The two formulas use the same coordinates but different operations — averages versus differences:

$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \qquad d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

A handy consequence: the distance from either endpoint to the midpoint is exactly $d/2$, which is a fast way to verify your answer.

Where it gets used

• Centre of a circle from the endpoints of a diameter.
• Perpendicular bisector: passes through $M$ with slope $-1/m$.
• Centroid of a triangle, and proving a quadrilateral is a parallelogram (its diagonals share a midpoint).

• Subtracting instead of averaging. The midpoint uses $x_1 + x_2$; the distance formula uses $x_2 - x_1$. Mixing them up is the single most frequent error, and it produces a point nowhere near the segment.
• Dividing only one coordinate by 2. Both (or all three) coordinates must be halved.
• Sign slips with negatives. For $(-3, 7)$ and $(5, -1)$: $(-3 + 5)/2 = 1$, not $-4$ or $4$. Write the addition out before simplifying.
• Using $M - A$ for a missing endpoint. That gives the half-vector, not the endpoint. The correct rule is $B = 2M - A$.
• Reporting the midpoint as a single number. It is an ordered pair (or triple), not a length.
• Assuming the midpoint splits the segment into equal slopes. Slope is constant along a straight segment; what the midpoint halves is length.