Midpoint Formula

The midpoint formula returns the point that lies exactly halfway along the straight line connecting two given points (x₁, y₁) and (x₂, y₂). It is a coordinate pair, not a single number, because the midpoint has both an x and a y component.

The midpoint is the geometric center of the segment. Both original points are the same distance from it, in opposite directions. If you drew the two points on a graph and connected them with a straight line, the midpoint would land exactly at the middle of that line.

Use the midpoint formula whenever you have two points expressed as coordinates and want to know what point sits exactly between them. Common situations:

• School geometry problems involving coordinate planes.
• Finding the center of a line segment for a graph or geometric proof.
• Game programming: spawning an object at the center of two characters.
• Computer graphics: subdivision and interpolation between two points.
• Engineering or physics: averaging two coordinate readings.

If your two points happen to lie on the same horizontal or vertical line, you do not strictly need the formula. Just average the differing coordinate. But the formula handles those cases correctly too, so there is no harm in always using it.

The formula is one of the simplest in coordinate geometry. It just averages each coordinate axis separately.

The formula spells out a simple recipe: average the x-coordinates, average the y-coordinates, and write the answer as an ordered pair. There is no Pythagorean theorem or square root involved, which is why the midpoint formula is much simpler than the distance formula.

The coordinates label the two points:

• x₁ and y₁ are the coordinates of the first point.
• x₂ and y₂ are the coordinates of the second point.
• (x₁, y₁) reads as “x sub 1, y sub 1.” The subscript just labels which point this is.

The numbering is purely a label. You can call either point “point 1” and the other “point 2,” and the midpoint formula gives the same answer either way. Adding x₁ + x₂ (or y₁ + y₂) is symmetric: order does not matter for addition.

The midpoint formula is just the average applied separately to each coordinate axis. To find the center of a segment, you find the average of the two endpoints in each direction.

Picture the segment from point 1 to point 2. The horizontal extent runs from x₁ to x₂. The exact middle of that range, the value halfway between them, is the average (x₁ + x₂) / 2. The same logic applies to the vertical extent: the y of the midpoint is (y₁ + y₂) / 2.

That is one-dimensional averaging applied twice: once to x and once to y. Combined into an ordered pair, it gives the geometric center of the segment.

Take the points (2, 4) and (8, 10).

1. Identify the coordinates: x₁ = 2, y₁ = 4, x₂ = 8, y₂ = 10.
2. Add the x coordinates: 2 + 8 = 10.
3. Divide by 2 for the x of the midpoint: 10 / 2 = 5.
4. Add the y coordinates: 4 + 10 = 14.
5. Divide by 2 for the y of the midpoint: 14 / 2 = 7.
6. Write the midpoint as an ordered pair: M = (5, 7).

The midpoint is (5, 7).

Take the points (0, 0) and (10, 10).

• x midpoint = (0 + 10) / 2 = 5
• y midpoint = (0 + 10) / 2 = 5
• M = (5, 5)

The midpoint is (5, 5), which makes sense: it is the center of the line from the origin to (10, 10).

Take the points (−4, 6) and (2, −2).

• x midpoint = (−4 + 2) / 2 = −2 / 2 = −1
• y midpoint = (6 + (−2)) / 2 = 4 / 2 = 2
• M = (−1, 2)

The midpoint is (−1, 2). The formula handles negative coordinates the same way as positive ones. Just be careful with the addition when negatives are involved.

Take the points (1.5, 2.5) and (4.5, 6.5).

• x midpoint = (1.5 + 4.5) / 2 = 6 / 2 = 3
• y midpoint = (2.5 + 6.5) / 2 = 9 / 2 = 4.5
• M = (3, 4.5)

Decimal coordinates work fine. The midpoint can also have decimals; there is no requirement that it land on a clean integer.

Both formulas take the same inputs (two points), but answer different questions:

• The midpoint formula gives the center point of the segment between them: an ordered pair (x, y).
• The distance formula gives the length of the segment between them: a single number.

The two formulas use the same coordinates but combine them differently:

• Midpoint: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
• Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)

Use whichever matches what you need. For more on the distance formula, see the Distance Formula guide for the article or the distance formula calculator for the tool.

A few traps that catch people:

• Subtracting instead of adding. The midpoint formula adds the coordinates, then divides by 2. Subtracting (which is the first step of the distance formula) gives a different answer.
• Forgetting to divide by 2. Adding the coordinates without dividing gives the sum, not the midpoint.
• Mixing up x and y. Average x with x and y with y. Do not cross-mix.
• Sign errors with negative coordinates. (−4 + 2) / 2 = −1, not −3 or 3. Add carefully.
• Writing the midpoint as a single number. The midpoint is an ordered pair (x, y), not a scalar. If the answer is just one number, something went wrong.

Solving by hand:

• Quick once you know the formula.
• Useful for understanding what the midpoint represents.
• Easy to slip on signs with negative coordinates.

Using a calculator:

• Fast: enter the four coordinates and read the answer.
• Handles negatives and decimals cleanly.
• Reports the breakdown (x midpoint, y midpoint) so you can see how the answer was constructed.

For the calculator approach, the midpoint formula calculator does the math in one step and shows the step-by-step breakdown using your actual numbers.

• Formula: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
• Average the x-coordinates for the x of the midpoint.
• Average the y-coordinates for the y of the midpoint.
• Write the result as an ordered pair (x, y).
• The midpoint can have decimals; the inputs can be negative.
• Midpoint gives the center point of the segment; the distance formula gives its length.
• The midpoint formula calculator runs the math in one step.