Midpoint Calculator

1. Pick 2D or 3D. Graph-paper problems are 2D. Points with a height or depth value are 3D.
2. Type the two endpoints A and B. Order does not matter for the midpoint. Negatives and decimals work; write fractions as decimals.
3. Read M, then the extras. The tool shows each averaging step, the length of AB, the half length, the slope and, in 2D, the perpendicular bisector and a plot.

The midpoint of a line segment is the one point that splits the segment into two equal parts. It lies on the segment itself, and its distance to each endpoint is half the segment length.

OpenStax College Algebra 2e defines the midpoint formula as the average of the endpoint coordinates. The idea extends to three dimensions: OpenStax Calculus Volume 3 finds the center of a sphere as the midpoint of a diameter by averaging x, y and z.

A midpoint is not the same as a centroid. The centroid of a triangle averages three vertices, while a midpoint averages two endpoints. It is also not the halfway point of a trip on the curved Earth, where latitude and longitude do not behave like flat coordinates.

The midpoint formula averages each coordinate on its own. An average of two numbers sits exactly between them, so averaging x and y separately puts M halfway along both axes at once.

1. Add the two x coordinates and divide by 2.
2. Add the two y coordinates and divide by 2. In 3D, do the same for z.
3. Write the results as the point M.
4. Find the length AB with the distance formula; AM and MB are each half of it.
5. For the perpendicular bisector, take the negative reciprocal of the slope of AB and pass the line through M.

Find the midpoint of the segment from A(-3, 5) to B(7, -1).

1. x: (-3 + 7) / 2 = 4 / 2 = 2.
2. y: (5 + (-1)) / 2 = 4 / 2 = 2.
3. So M = (2, 2).
4. Length: the changes are 10 and -6, so AB = sqrt(100 + 36) = sqrt(136) = 11.661904, and AM = MB = 5.830952.

The slope of AB is -6 / 10 = -0.6. The perpendicular slope is 10 / 6 = 1.666667, so the perpendicular bisector through (2, 2) is y = 1.666667x - 1.333333.

A 3D Example from a Sphere

OpenStax Calculus Volume 3 uses the diameter from P(-5, 2, 3) to Q(3, 4, -1). The center is ((-5 + 3) / 2, (2 + 4) / 2, (3 + (-1)) / 2) = (-1, 3, 1). The diameter is sqrt(84) = 9.165151, so the radius is 4.582576.

The midpoint depends only on the two endpoints, but signs, dimensions and scale all affect how you read it.

Negative Coordinates

Adding a negative number is where most slips happen. (5 + (-1)) / 2 is 2, not 3. The method line wraps negative values in brackets.

Number of Dimensions

A 3D point has three coordinates, so the midpoint has three averages. Leaving out z places the answer on the floor of the space instead of at the true center.

Axis Scale

The formula assumes the same unit on every axis. A chart with years on x and dollars on y still has a midpoint, but its distance and perpendicular bisector have no physical meaning.

Vertical and Horizontal Segments

A vertical segment has an undefined slope, so its perpendicular bisector is horizontal: y = constant. A horizontal segment gives a vertical bisector: x = constant.

These three segment tools answer different questions from the same kind of input.

When the midpoint is known and one end is missing, run the formula in reverse with the endpoint calculator.

Coordinate Geometry Homework

Midpoint questions appear in algebra, geometry and precalculus. The step layout shows each average, which is what graders look for.

Circle Centers

The center of a circle is the midpoint of any diameter, and the radius is the half distance. From there the circle calculator gives the area and circumference.

Perpendicular Bisectors

Every point on the perpendicular bisector is the same distance from A and B. Constructions, Voronoi-style regions and circumcenters all start here. The slope calculator explains the negative reciprocal step.

Design and Layout on a Grid

Centering an object between two marks on a floor plan or a screen uses the same average. Pixel and CAD coordinates are flat, so the formula applies directly.

1. Subtracting Instead of Adding

(x2 - x1) / 2 is the half step, not the midpoint. The midpoint adds the coordinates first.

2. Forgetting to Divide by 2

x1 + x2 alone is twice the midpoint coordinate. For A(-3, 5) and B(7, -1) it gives (4, 4) instead of (2, 2).

3. Mixing X and Y Values

Average x with x and y with y. Pairing x1 with y2 produces a point that is not on the segment.

4. Dropping a Sign

Write negatives in brackets before adding. -3 + 7 is 4, not -10.

5. Averaging Latitude and Longitude

Map coordinates sit on a sphere. The plain average drifts from the true halfway point on long routes, so use a great-circle method.

6. Using the Wrong Slope for the Bisector

The bisector slope is the negative reciprocal, not the negative. For slope -0.6 it is 1.666667, not 0.6. To write the full line in another form, the linear equation solver handles the algebra.

The midpoint is exact for the numbers you enter; results show up to 6 decimal places.

No. (x1 + x2) / 2 equals (x2 + x1) / 2, so swapping A and B gives the same midpoint.

Yes. The midpoint of (0, 0) and (3, 1) is (1.5, 0.5). An odd sum always gives a half.

It is half of each coordinate. The midpoint of (8, -6) and (0, 0) is (4, -3).

Three points have no single midpoint. Their average, ((x1 + x2 + x3) / 3, ...), is the centroid of the triangle they form.

Yes. It lies on the straight segment AB, exactly half the length from each end.

Take the midpoint of A and M. For A(-3, 5) and M(2, 2) that gives (-0.5, 3.5).

The center is the midpoint of any diameter. The radius is half the diameter length.

Map apps work on a sphere. Averaging latitude and longitude only approximates the halfway point, and the error grows with distance.

Yes. Use 2D with both y values set to 0. The midpoint of 3 and 11 is 7.

No. The math runs in your browser. Save keeps a copy only in your own browser storage.

• The midpoint formula for two points in the plane (OpenStax College Algebra 2e, section 2.1).
• Distance and midpoint formulas with circles (OpenStax Intermediate Algebra 2e, section 11.1).
• Distance and the midpoint of a diameter in three dimensions (OpenStax Calculus Volume 3, section 2.2).