Midpoint Calculator
The point halfway between two coordinates, or the missing endpoint.
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- x coordinate—
- y coordinate—
- z coordinate—
- Distance between the points—
- Half the distance—
- Slope of the line—
How to use this midpoint calculator
- Pick what you are solving for. Either the midpoint between two known points, or a missing endpoint when you already know one end and the middle.
- Choose two or three dimensions and enter the coordinates.
- Read the answer, along with the distance between the points and the slope of the line joining them.
The midpoint formula
The midpoint of a line segment is the point exactly halfway along it. You find it by averaging the coordinates:
In three dimensions, average the z values as well. That is the entire method. Unlike the distance formula there is nothing squared and no square root, which makes this one of the few coordinate formulas that is genuinely easy to do in your head.
The logic is plain once stated: the halfway point in the x direction is the average of the two x values, and the same is true independently for y and z. Averaging each axis separately gives the point at the centre.
A worked example
Find the midpoint of A(2, 3) and B(8, 11).
- Average the x values. (2 + 8) ÷ 2 = 5
- Average the y values. (3 + 11) ÷ 2 = 7
- Write the result. The midpoint is (5, 7).
Check it by measuring: from A to M is 3 across and 4 up, and from M to B is also 3 across and 4 up. Both halves are identical, which is what being the midpoint means.
Working backwards to a missing endpoint
A common exam question reverses the problem: you know one endpoint and the midpoint, and need the other endpoint. Rearranging the formula gives:
In words: double the midpoint, then subtract the endpoint you already have. If A is (2, 3) and the midpoint is (5, 7), then B is (10 − 2, 14 − 3), which is (8, 11).
The instinct is usually to subtract A from M and add the difference again, which works but takes longer. Doubling the midpoint is the same operation in one step. Switch the mode above and the calculator does it this way.
Midpoint, distance and slope
| Quantity | Formula | Operation |
|---|---|---|
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) | Add and halve |
| Distance | √(Δx² + Δy²) | Subtract, square, root |
| Slope | Δy / Δx | Subtract and divide |
Note the pattern: the midpoint adds the coordinates while distance and slope subtract them. Mixing these up is the single most common error with these formulas. If you find yourself subtracting while hunting for a midpoint, you have reached for the wrong one.
What the midpoint is used for
- Perpendicular bisectors. The line that cuts a segment in half at right angles must pass through the midpoint, so finding it is always the first step.
- Centre of a circle. Given the two ends of a diameter, the centre is their midpoint.
- Proving shapes. A quadrilateral is a parallelogram exactly when its two diagonals share a midpoint, which turns a geometry proof into two quick calculations.
- Design and layout. Centring an element between two fixed points is a midpoint calculation, whether done by hand or by a layout engine.
- Binary search. The algorithm repeatedly takes the midpoint of a range to halve the search space, which is why it finds an item among a million in about twenty steps.
- Animation and interpolation. Finding a point partway between two positions generalises the midpoint: halfway is the special case where the fraction is 0.5.
Beyond halfway
The midpoint is one case of a more general idea. To find the point a fraction t of the way from A to B, use x = x₁ + t(x₂ − x₁). Setting t to 0.5 gives the midpoint; setting it to 0.25 gives the quarter point. This is linear interpolation, and it is how every animation moves an object smoothly between two positions.
Frequently asked questions
What is the midpoint formula?
The midpoint is the average of the coordinates: M = ((x1 + x2) / 2, (y1 + y2) / 2). In three dimensions you average the z values too. Add the pairs and halve each one.
What is the midpoint of (2, 3) and (8, 11)?
It is (5, 7). The x values average to (2 + 8) / 2 = 5 and the y values average to (3 + 11) / 2 = 7.
How do I find a missing endpoint from the midpoint?
Double the midpoint and subtract the endpoint you know: B = 2M – A. If A is (2, 3) and the midpoint is (5, 7), then B is (10 – 2, 14 – 3), which gives (8, 11).
Is the midpoint the same as the average?
Yes. The midpoint is simply the average of each coordinate taken separately. Averaging the x values gives the midpoint’s x, and averaging the y values gives its y.
Can the midpoint have decimal or negative coordinates?
Yes. Averaging two whole numbers of different parity gives a half, so a midpoint of (2.5, -1.5) is perfectly normal. Negative coordinates work the same way as positive ones.
What is the difference between the midpoint and distance formulas?
The midpoint adds the coordinates and halves them, giving a point. The distance formula subtracts them, squares, adds and takes a square root, giving a length. Confusing add with subtract is the most common mistake.
How do I find the midpoint in three dimensions?
Average all three coordinates: M = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2). Switch this calculator to three dimensions and the z boxes appear.
Why does the midpoint matter in geometry proofs?
Because several shape properties depend on it. A quadrilateral is a parallelogram exactly when its diagonals share a midpoint, and a perpendicular bisector must pass through the midpoint of the segment it cuts.