Midpoint Calculator
Find the point halfway between two points by averaging their coordinates, with the steps and the length of each half of the segment.
What is a midpoint calculator?
The midpoint of a line segment is the point exactly halfway between its two endpoints. It lies on the segment and is the same distance from both ends, so it splits the segment into two equal halves. In coordinates, finding it is simple: average the two x-coordinates and average the two y-coordinates.
This midpoint calculator takes two endpoints, (x₁, y₁) and (x₂, y₂), and returns the Midpoint (x, y) as a coordinate pair, along with Midpoint x and Midpoint y as separate numbers. The Step-by-step line shows each average worked out with your numbers, and the Half-length line gives the length of each half of the segment, which is half the distance between the two points. That last figure is useful when the segment is the diameter of a circle, because the half-length is then the radius.
In the Common Core high school geometry standards, the midpoint is the simplest case of finding the point that partitions a directed segment in a given ratio (G-GPE.B.6), here the ratio 1 : 1, and coordinates are used to prove simple geometric theorems algebraically (G-GPE.B.4). For example, proving that the diagonals of a parallelogram bisect each other comes down to showing that both diagonals have the same midpoint.
Formula
For endpoints (x₁, y₁) and (x₂, y₂):
M = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )
Each coordinate of the midpoint is the average of the endpoints' coordinates. An equivalent way to see it: start at the first point and go half of the way across, M = (x₁ + Δx ÷ 2, y₁ + Δy ÷ 2), where Δx = x₂ − x₁ and Δy = y₂ − y₁. Both forms give the same point, and swapping the endpoints does not change the answer, because addition does not depend on order.
Half-length
Each half of the segment has length
½ × √((x₂ − x₁)² + (y₂ − y₁)²)
which is half of the distance formula. The calculator rounds it to six decimal places.
Working backward: a missing endpoint
If you know the midpoint M = (m₁, m₂) and one endpoint (x₁, y₁), solve the midpoint formula for the other endpoint:
(x₂, y₂) = (2m₁ − x₁, 2m₂ − y₁)
The calculator does not do this step for you, but you can check the result by entering both endpoints and confirming that it returns your midpoint.
Related constructions
- Perpendicular bisector: the line through the midpoint with the negative reciprocal of the segment's slope. Every point on it is the same distance from both endpoints.
- Other ratios: a point one-quarter or two-thirds of the way along the segment comes from the section formula, which generalizes the midpoint formula. See the section formula explained.
Worked example
Example 1: negative coordinates
Find the midpoint of the segment from (−3, 4) to (6, −2).
- x: (−3 + 6) ÷ 2 = 3 ÷ 2 = 1.5.
- y: (4 + (−2)) ÷ 2 = 2 ÷ 2 = 1.
- Midpoint: (1.5, 1).
The step line reads x = (-3 + 6) ÷ 2 = 3 ÷ 2 = 1.5, y = (4 + (-2)) ÷ 2 = 2 ÷ 2 = 1, and the half-length line reads Each half of the segment is 5.408327 long. The whole segment is twice that, √117 = 3√13 ≈ 10.816654, which the distance formula confirms.
Example 2: the center and radius of a circle
The endpoints of a diameter are (−4, −1) and (2, 7). The center of the circle is the midpoint of the diameter.
- x: (−4 + 2) ÷ 2 = −2 ÷ 2 = −1. y: (−1 + 7) ÷ 2 = 6 ÷ 2 = 3.
- Center: (−1, 3).
- The half-length is 5, so the radius is 5.
The circle's equation is therefore (x + 1)² + (y − 3)² = 25.
Example 3: a missing endpoint
M = (4, 1) is the midpoint of a segment with one endpoint at (1, −3). Find the other endpoint.
- x₂ = 2 × 4 − 1 = 7.
- y₂ = 2 × 1 − (−3) = 2 + 3 = 5.
- The other endpoint is (7, 5).
Check: entering (1, −3) and (7, 5) returns a midpoint of (4, 1), with steps x = (1 + 7) ÷ 2 = 8 ÷ 2 = 4, y = (-3 + 5) ÷ 2 = 2 ÷ 2 = 1 and a half-length of 5. A common wrong answer is to average the midpoint and the known endpoint, which gives (2.5, −1). That point is the midpoint of the first half, one-quarter of the way along the segment, not the far end.
How to use this calculator
- Enter the first endpoint in x₁ and y₁.
- Enter the second endpoint in x₂ and y₂. Negative numbers and decimals are fine.
- Read the Midpoint (x, y). Use Midpoint x and Midpoint y if you need the coordinates separately.
- Compare the Step-by-step line with your working.
- Use the Half-length when you need the distance from the midpoint to either end, such as the radius of a circle.
Common mistakes
- Subtracting instead of adding. The midpoint uses x₁ + x₂; the difference x₂ − x₁ belongs to the slope and distance formulas.
- Forgetting to divide by 2, or dividing only one coordinate.
- Mixing coordinates, such as adding x₁ to y₂.
- Sign errors with negatives. 4 + (−2) is 2, not 6.
- Averaging the wrong pair when looking for a missing endpoint, as in Example 3.
What the tool does and does not do
- It finds the midpoint of two points in the plane. For three dimensions, average the z-coordinates the same way.
- It does not solve for a missing endpoint or divide a segment in other ratios; use the formulas above and check the result with the calculator.
- Results are rounded to six decimal places.
For more uses of the formula, read where the midpoint formula is used in geometry. For the full length of the segment, use the distance formula calculator; for its slope and the slope of its perpendicular bisector, use the slope calculator.
Sources
Frequently asked questions
Is the midpoint the same as the average?
Yes, coordinate by coordinate. The x-coordinate of the midpoint is the mean of the two x-coordinates, and the y-coordinate is the mean of the two y-coordinates. On a number line, the midpoint of 2 and 8 is 5, their average; the midpoint formula does the same thing in each direction. That is also why it extends to three dimensions: the midpoint of (1, 2, 3) and (5, 6, 9) is (3, 4, 6), with the z-coordinates averaged too. This calculator takes two coordinates per point, so average z by hand.
Can I enter decimals or fractions?
Decimals work directly: (2.5, −1.5) and (4, 3) give a midpoint of (3.25, 0.75) and a half-length of 2.371708. Enter a fraction as a decimal. A fraction whose decimal does not end, such as 1/3, can only be typed approximately (0.333333), and the results are rounded to six decimal places, so for an exact answer add the fractions by hand: the midpoint of 1/3 and 1/2 on the x-axis is (1/3 + 1/2) ÷ 2 = (5/6) ÷ 2 = 5/12.
Does it matter which endpoint I enter first?
No. Addition gives the same sum in either order, so (−3, 4) and (6, −2) give (1.5, 1) whichever you enter first. Only the order of the numbers in the step line changes.
How do I find an endpoint when I know the midpoint?
Double the midpoint and subtract the known endpoint, one coordinate at a time: (x₂, y₂) = (2m₁ − x₁, 2m₂ − y₁). With midpoint (4, 1) and endpoint (1, −3), the other endpoint is (7, 5), as in Example 3. Enter both endpoints in the calculator to confirm that the midpoint comes out as (4, 1).
What does the half-length tell me?
It is the distance from the midpoint to either endpoint, which is half the length of the whole segment. For a diameter it is the radius. For (1, −3) and (7, 5), the half-length is 5, so the full segment is 10 units long.
How do I find the perpendicular bisector of a segment?
You need the midpoint and the negative reciprocal of the segment's slope. For the segment from (1, −3) to (7, 5), the midpoint is (4, 1) and the slope is 8 ÷ 6 = 4/3, so the perpendicular slope is −0.75. The bisector is y − 1 = −0.75(x − 4), or y = −0.75x + 4. The point (8, −2) lies on it, and the distance formula gives 5√2 from it to each endpoint.
How do I find a point one-quarter of the way along a segment?
One way is to take the midpoint twice. The midpoint of (1, −3) and (7, 5) is (4, 1); the midpoint of (1, −3) and (4, 1) is (2.5, −1), which is one-quarter of the way from (1, −3) to (7, 5). For any other ratio, use the section formula, which generalizes the midpoint formula.
Is a point that is equally far from both endpoints always the midpoint?
Not always. Every point on the perpendicular bisector is the same distance from both endpoints, such as (8, −2) in the bisector example. The midpoint is the only one of those points that is also on the segment itself.