Geometry
Midpoint Calculator
Find the exact centre between two points — in 2D or 3D — or work backwards from a midpoint to a missing endpoint. The result, breakdown, and coordinate plot update live as you type.
Enter your two points
Results update as you type. No Calculate button.
Point A
Point B
Midpoint M
(5, 5)
By coordinate
- x-axis(2 + 8) ÷ 2 = 5
- y-axis(3 + 7) ÷ 2 = 5
Midpoint shown in accent
Distance A↔B
7.2111
Midpoint lies in
Quadrant I
Updates live
Try an example
No sign-up · Free
The basics
What is the midpoint?
The midpoint is the point that sits exactly halfway along the straight segment joining two points — the same distance from each end. Finding it is just averaging: take the mean of the two x-coordinates, then the mean of the two y-coordinates. Because each axis is handled on its own, the idea extends cleanly from a number line to 2D and 3D.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
It's an average, per axis
Treat x and y separately. The midpoint's x is the average of the two x-values; its y is the average of the two y-values. Nothing couples the axes together.
(2, 8) → x = (2 + 8)/2 = 5
MA = MB
Equidistant from both ends
The midpoint is the unique point on the segment that is the same distance from each endpoint. It splits the segment into two equal halves, which is why it is also called the segment's centre.
halfway: same length to A as to B
Method
How to find a midpoint by hand
Three short steps — the same whether the numbers are whole, negative, or decimal.
- 1
Label your two points
Name one point A = (x₁, y₁) and the other B = (x₂, y₂). It doesn't matter which is which — the midpoint is the same either way.
A = (2, 3) B = (8, 7) - 2
Average the x-coordinates
Add the two x-values and divide by two. That gives the x-coordinate of the midpoint.
(2 + 8) ÷ 2 = 5 - 3
Average the y-coordinates
Do the same for the two y-values. Pair the results to get the midpoint as an ordered point.
(3 + 7) ÷ 2 = 5 → M = (5, 5)
Worked examples
See it across cases
The same averaging idea handles 2D points, 3D points, and the reverse problem of recovering a missing endpoint. Switch tabs to watch the working change.
Given A = (2, 3) and B = (8, 7)
Average each axis:
x: (2 + 8) ÷ 2 = 5
y: (3 + 7) ÷ 2 = 5
M = (5, 5)
The midpoint of the segment.
Reference
Every form of the formula
One idea, written for each dimension and for the reverse problem. Each averages the matching coordinates — except the endpoint form, which inverts the average.
| Variant | Formula | When to use it |
|---|---|---|
| 2D midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2) | Points on a flat coordinate plane |
| 3D midpoint | ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2) | Points in space, with a z-axis |
| On a number line | (a + b)/2 | Midpoint of two values in 1D |
| Missing endpoint | B = 2M − A | When the midpoint is known, one end isn't |
Where it shows up
Why the midpoint matters
A small average that anchors a surprising amount of geometry, design, and everyday measurement.
perpendicular bisector
Geometry & proofs
The midpoint is the foot of the perpendicular bisector and the centre of a circle drawn on a diameter — the starting point for countless constructions and proofs.
centre of a screen
Design & layout
Centring an element, placing a label between two markers, or finding the middle of a bounding box all reduce to averaging two corner positions.
map waypoints
Maps & navigation
The halfway point between two locations — a meeting spot or a route checkpoint — is the midpoint of their coordinates on the grid.
Worth knowing
Rules worth remembering
A handful of properties that make the midpoint quick to reason about.
order doesn't matter
Symmetric in A and B
Swapping the two points gives the same midpoint, because addition is commutative. There is no first or second point.
B = 2M − A
Reversible
Knowing the midpoint and one endpoint pins down the other exactly. Double the midpoint and subtract the known end.
half of a half
Always on the segment
The midpoint always lies on the segment between the two points — never outside it — and divides it in a perfect one-to-one ratio.
x₁ + x₂ odd → .5
Can be fractional
If the two coordinates on an axis have different parity, that axis of the midpoint lands on a half. A non-integer midpoint is normal.
same axes, per axis
Scales by dimension
Add a z-axis and the recipe is unchanged — average each new coordinate. The formula grows one term at a time.
centroid of two
It's the mean point
The midpoint is exactly the centroid (centre of mass) of two equal points — the two-point case of averaging a whole set of points.
Quick chart
Midpoints of common point pairs
The midpoint and segment length for frequently searched point pairs — handy for a fast check or homework spot-test.
| Point A | Point B | Midpoint M | Distance |
|---|---|---|---|
| (0, 0) | (4, 6) | (2, 3) | 7.2111 |
| (0, 0) | (10, 10) | (5, 5) | 14.1421 |
| (1, 1) | (5, 5) | (3, 3) | 5.6569 |
| (−2, −3) | (4, 5) | (1, 1) | 10 |
| (2, 3) | (8, 7) | (5, 5) | 7.2111 |
| (−4, 2) | (6, −8) | (1, −3) | 14.1421 |
| (0, 0) | (−6, 8) | (−3, 4) | 10 |
| (3, −5) | (−7, 9) | (−2, 2) | 17.2047 |
| (−1, −1) | (9, 7) | (4, 3) | 12.8062 |
| (5, 0) | (0, 5) | (2.5, 2.5) | 7.0711 |
| (−3, −4) | (3, 4) | (0, 0) | 10 |
| (12, 4) | (−2, −10) | (5, −3) | 19.799 |
Pitfalls
Common mistakes to avoid
The midpoint is simple, but a few slips catch people out.
subtract, don't
Subtracting instead of adding
The midpoint adds the coordinates and divides by two. Subtracting is the start of the distance formula, not the midpoint.
mind the sign
Dropping a negative
Keep the sign when a coordinate is negative. Adding a negative reduces the sum — losing the minus shifts the midpoint to the wrong place.
(x with x)
Mixing the axes
Average x with x and y with y. Pairing an x-value with a y-value scrambles the point entirely.
2M − A, not M − A
Forgetting the factor of two
To recover an endpoint, double the midpoint first: B = 2M − A. Just subtracting A from M is a classic off-by-half error.
Background
Where the formula comes from
The midpoint formula is a direct consequence of placing geometry on a coordinate grid — the analytic geometry pioneered by René Descartes in the 17th century. Once points became pairs of numbers, the "middle" of a segment became the average of those numbers, and the idea of section (dividing a segment in a given ratio) generalised it: the midpoint is simply the 1:1 section point. Today it is a building block for the distance formula, perpendicular bisectors, and the centroid of a triangle, which is the average of its three vertices.
FAQ
Common questions
The midpoint of two points (x₁, y₁) and (x₂, y₂) is M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). You average the x-coordinates to get the midpoint's x, and average the y-coordinates to get its y. It is the exact centre of the line segment joining the two points.
Add the two x-coordinates and divide by 2, then add the two y-coordinates and divide by 2. For (2, 3) and (8, 7): x is (2 + 8) ÷ 2 = 5 and y is (3 + 7) ÷ 2 = 5, so the midpoint is (5, 5).
In three dimensions you average all three coordinates: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2, (z₁ + z₂) ÷ 2). The method is identical to 2D — switch the tool to the 3D mode to enter a z value for each point.
Rearrange the formula to B = 2M − A. Double each coordinate of the midpoint and subtract the matching coordinate of the known endpoint. If A is (−2, 4) and M is (3, 1), then B is (2·3 − (−2), 2·1 − 4) = (8, −2). The 'Find endpoint' mode does this for you.
Yes. The midpoint is exactly the arithmetic mean of the two points taken coordinate by coordinate. Averaging the x-values and the y-values separately is the same as finding the point halfway along the segment.
Yes. The formula handles negatives, decimals, and fractions without any change — just keep the signs. For (−4, 2) and (6, −8) the midpoint is ((−4 + 6) ÷ 2, (2 + −8) ÷ 2) = (1, −3).
The midpoint formula averages coordinates to find the centre point and returns a point. The distance formula uses the Pythagorean theorem to measure the length between two points and returns a single number. They share the same two inputs but answer different questions.
Often, yes. Because you divide a sum by 2, the midpoint lands on a half whenever the two coordinates have different parity. The midpoint of (1, 1) and (4, 6) is (2.5, 3.5) — perfectly valid, just not a whole-number grid point.
Keep going
Related calculators
Keep reading
Further reading
Math
Slope-Intercept Form (y = mx + b): How to Find and Graph It
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This guide shows how to find the equation from a graph, two points, or a point and slope, graph it, and convert standard form.
Ugo Charles
Math
The Cylinder Volume Formula (V = πr²h): How It Works, With Examples
The volume of a cylinder is V = πr²h, where r is the radius of the circular base, h is the height, and π ≈ 3.14159. This guide shows why the formula works, walks through a full numerical example, and covers diameter inputs and unit conversions.
Ugo Charles
Math
The Distance Formula: How to Find the Distance Between Two Points
The distance between two points is d = √((x₂−x₁)² + (y₂−y₁)²). It comes straight from the Pythagorean theorem: the horizontal and vertical gaps between the points are the two legs of a right triangle, and the distance is the hypotenuse.
Ugo Charles
Written by
UC
Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
Did this solve your problem?