Midpoint Calculator

About the Midpoint Calculator

The midpoint of a segment is the point exactly halfway between its ends. It appears throughout coordinate geometry: finding the centre of a circle from a diameter, constructing perpendicular bisectors, proving that the diagonals of a parallelogram bisect each other or locating the middle of a route on a map grid.

The formula itself is short, but sign errors with negative coordinates and fractions are common. This calculator averages each coordinate exactly, keeps fractions as fractions, and then checks the result by showing that the midpoint is the same distance from both ends. It also runs the formula in reverse, which is a frequent exam question: given one endpoint and the midpoint, where is the other endpoint?

What is the midpoint formula?

For points $A(x_1, y_1)$ and $B(x_2, y_2)$ , the midpoint is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ . Each coordinate of the midpoint is simply the average of the corresponding coordinates of the endpoints. In three dimensions, the z-coordinates are averaged in the same way.

Averaging works because moving from A to B changes x by $x_2 - x_1$ , and halfway along that change is $x_1 + \frac12(x_2 - x_1) = \frac{x_1 + x_2}{2}$ . The same reasoning applies to every coordinate independently.

How to use the Midpoint Calculator

Choose whether you want the midpoint or a missing endpoint.
Enter the first point as (x, y) or (x, y, z).
Enter the second point, or the midpoint if you are finding an endpoint.
Press Find point.
Read the answer and the distance check.

Formulas

Midpoint in the plane

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

Midpoint in space

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

Missing endpoint

B = 2M - A

Distance check

|AM| = |MB| = \tfrac12 |AB|

Worked example: A(1, 2), B(7, −4)

The worked example finds the midpoint of $A(1, 2)$ and $B(7, -4)$ . The x-coordinate is $\frac{1 + 7}{2} = 4$ and the y-coordinate is $\frac{2 + (-4)}{2} = -1$ , so $M = (4, -1)$ .

As a check, both $|AM|$ and $|MB|$ equal $\sqrt{3^2 + 3^2} = 3\sqrt2 \approx 4.243$ , which is half of $|AB| = 6\sqrt2 \approx 8.485$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

M = (4, -1)

How we read your problem

The midpoint is the point halfway along the segment AB.

A = (1, 2),\ B = (7, -4)

Step-by-step solution

Average the x-coordinates
$x_M = \frac{1 + 7}{2} = 4$
Average the y-coordinates
$y_M = \frac{2 + -4}{2} = -1$
Check: M is the same distance from both ends
$|AM| = |MB| = 3\sqrt{2} \approx 4.242640687,\quad |AB| = 6\sqrt{2} \approx 8.485281374$

Formulas used

Midpoint
$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Missing endpoint
$B = 2M - A$

Each coordinate of the midpoint is the average of the endpoints' coordinates, so the midpoint splits the segment into two equal halves.

Verification

|AM| = |MB| = 3√2 ≈ 4.242640687, half of |AB| = 6√2 ≈ 8.485281374.

Detailed explanation

Finding a missing endpoint

If M is the midpoint of AB, then $M = \frac{A + B}{2}$ , so $B = 2M - A$ . For $A(2, 3)$ and $M(5, -1)$ , the missing endpoint is $B = (2 \times 5 - 2, 2 \times (-1) - 3) = (8, -5)$ .

A quick way to see this is to note how far A is from M in each direction and step the same amount past M. From $x = 2$ to $x = 5$ is $+3$ , so B has $x = 8$ ; from $y = 3$ to $y = -1$ is $-4$ , so B has $y = -5$ .

Where midpoints are used

The centre of a circle is the midpoint of any diameter, so if a diameter runs from $(-3, 1)$ to $(5, 7)$ , the centre is $(1, 4)$ and the radius is half the length of the diameter, here 5. The perpendicular bisector of a segment passes through its midpoint and is perpendicular to it, which is how the circumcentre of a triangle is found.

In proofs, midpoints help show properties of shapes. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long, and the diagonals of a parallelogram share a common midpoint.

Fractions and three dimensions

When the sum of coordinates is odd, the midpoint has a half in it. The calculator keeps such values exact; the midpoint of $(\frac12, 3, 0)$ and $(2, -1, 5)$ is $(\frac54, 1, \frac52)$ . Exact values are preferable when the midpoint feeds into further calculations, such as a slope or an equation of a line.

Three-dimensional problems use the same averaging, with one extra coordinate. Both points must have the same number of coordinates; mixing a 2D point with a 3D point is reported as an input error.

Checking the answer with distances

A quick way to confirm a midpoint is to measure. The distance from A to M must equal the distance from M to B, and each must be exactly half of the full length AB. For the diameter from (−3, 1) to (5, 7), the centre is (1, 4); each radius has length $\sqrt{16 + 9} = 5$ , and the whole diameter has length 10.

The calculator reports the segment length alongside the point so you can run this check yourself. If the two halves ever come out different, one of the coordinates was copied or added incorrectly, most often a sign on a negative value.

Common mistakes

Subtracting instead of adding the coordinates before halving.
Dropping a negative sign when adding coordinates.
Using the midpoint formula to find a missing endpoint without rearranging it.
Averaging x with y instead of x with x.
Confusing the midpoint with the distance between the points.

Tips

Write both points one above the other so matching coordinates line up.
Plot the points roughly; the midpoint should sit visibly in the middle.
For a missing endpoint, think "step past the midpoint by the same amount".
Use the distance formula calculator to find the length of the segment.
Use the slope calculator with the midpoint to build a perpendicular bisector.

Frequently asked questions

How do you find the midpoint between two points?

Add the x-coordinates and divide by 2, then add the y-coordinates and divide by 2. The two averages are the coordinates of the midpoint.

How do I find an endpoint if I know the midpoint?

Use B = 2M − A: double each coordinate of the midpoint and subtract the matching coordinate of the known endpoint.

Does the midpoint formula work in three dimensions?

Yes. Average the z-coordinates as well: M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2).

Can a midpoint have fractional coordinates?

Yes. Whenever two matching coordinates add to an odd number, their average ends in one half. The calculator keeps these as exact fractions.

How is the midpoint related to the centre of a circle?

The centre of a circle is the midpoint of any diameter. Find the midpoint of the diameter's endpoints to locate the centre.

Is the midpoint the same as the average of two points?

Yes. The midpoint is the coordinate-wise average of the two points, which is why the formula divides each sum by 2.

Find your midpoint

Enter two points above to get the exact midpoint, or an endpoint and the midpoint to find the missing end.

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