Midpoint Calculator

What is a midpoint?

The midpoint of a line segment is the one point that lies exactly halfway between its two end points. It cuts the segment into two pieces of equal length, which is why drawing a midpoint is also called bisecting the segment. Every segment in the plane has exactly one midpoint, and it always lies on the segment itself.

You do not need angles or lengths to find it. Each coordinate of the midpoint is the middle value of the two matching coordinates of the end points, so the whole job comes down to two averages.

Midpoint formula

For a segment from A = (x₁, y₁) to B = (x₂, y₂), add the two x values and halve the sum, then do the same with the y values.

M = ((x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2)The x and y coordinates are handled independently of each other.

Negative coordinates need no special treatment: keep the signs when you add. Swapping A and B changes nothing, because addition does not care about order. The same idea extends to three dimensions by averaging a third coordinate, z, in the same way; this calculator covers the flat, two-coordinate case.

How to use the calculator

1. Enter the x and y coordinates of point A in the first row.
2. Enter the x and y coordinates of point B in the second row.
3. Press Calculate midpoint to see M as an (x, y) pair.

Answers are rounded to four decimal places. Whole numbers appear without decimals, and values that need only one or two decimals are padded to two, so 1.5 is shown as 1.50. Both coordinates of each point are required, and plain decimal numbers such as -3.5 or 2.25 are accepted.

Worked examples

Example 1. Find the midpoint of A = (−3, 4) and B = (7, −2). For x: (−3 + 7) ÷ 2 = 4 ÷ 2 = 2. For y: (4 + (−2)) ÷ 2 = 2 ÷ 2 = 1. The midpoint is M = (2, 1).

To check, measure the distances with the Pythagorean theorem. From A to B the run is 10 and the rise is −6, so the length is √(10² + 6²) = √136 ≈ 11.6619. From A to M the run is 5 and the rise is −3, giving √34 ≈ 5.8310, which is exactly half of the full length.

Example 2. Take A = (1.5, 2.25) and B = (4, −3.5). The x coordinate is (1.5 + 4) ÷ 2 = 2.75 and the y coordinate is (2.25 − 3.5) ÷ 2 = −0.625. The calculator displays (2.75, -0.6250), using four decimals because the y value needs three.

Integer end points can still give a fractional midpoint: for (1, 2) and (4, 7) the sums are 5 and 9, so M = (2.5, 4.5).

Finding a missing end point

Sometimes the midpoint M and one end A are known and the other end B is the unknown. Rearranging the formula gives B by doubling the midpoint and subtracting the known end.

B = (2·xM − x₁ , 2·yM − y₁)

With M = (2, 1) and A = (−3, 4) this gives B = (2·2 + 3, 2·1 − 4) = (7, −2), the same segment as in Example 1. The endpoint calculator does this step for you.

Midpoints in geometry

Midpoints are the starting point for many classic constructions and theorems:

• Perpendicular bisector. The line that crosses a segment at its midpoint at a right angle. Every point on it is the same distance from both ends.
• Medians of a triangle. A median joins a vertex to the midpoint of the opposite side. The three medians meet at the triangle's centroid. See the area of a triangle and perimeter of a triangle calculators for other triangle measures.
• Midsegment theorem. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long.
• Circles. If a segment is a diameter, its midpoint is the centre of the circle.
• Parallelograms. The two diagonals bisect each other, so they share a single midpoint.

Other places a midpoint appears

The midpoint formula is the average of two values, so it shows up well outside geometry. Two equal masses balance at the midpoint between them. In statistics the midpoint of a class interval, such as 20 to 30, is the single number 25 used to represent that group. In graphics and layout work, the midpoint of two corners gives the spot where a label or handle should be centred.

Averaging latitude and longitude is only a rough shortcut over short distances. On long routes the Earth's curvature means the true halfway point lies along a great-circle arc, not at the plain average of the coordinates.

Common mistakes

• Adding the coordinates but forgetting to divide by two, which gives the sum of the points rather than the middle.
• Averaging an x value with a y value instead of matching x with x and y with y.
• Dropping a minus sign when one of the points lies left of or below the origin.