How to use the Midpoint Calculator
- Enter the first endpoint as x₁ and y₁.
- Enter the second endpoint as x₂ and y₂.
- Read the midpoint coordinates, given as an ordered pair.
- Use the segment length when you need the span as well as its centre.
- Take the perpendicular bisector slope when constructing a line that cuts the segment at right angles.
How the calculation works
The midpoint is the arithmetic mean of the coordinates, taken one axis at a time. Averaging the x values gives the horizontal centre and averaging the y values gives the vertical centre; together they locate the point exactly halfway along the segment. This works because a straight segment is parameterised linearly, so the halfway point in distance is also the halfway point in each coordinate.
The perpendicular bisector is the set of points equidistant from both endpoints. It passes through the midpoint and has slope equal to the negative reciprocal of the segment's slope. That construction underpins a surprising range of practical work: finding the centre of a circle through three points, locating the equidistant boundary between two service depots, and building Voronoi diagrams that partition territory by nearest facility.
Because the calculation is a plain average, it extends without modification. A three-dimensional midpoint averages z as well; the centroid of several points averages every coordinate across all of them. What it does not do is find a point of balance when the endpoints carry different weights — for that you need a weighted average, where each coordinate is multiplied by its weight before summing and dividing by the total weight.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2); perpendicular slope = −(x₂ − x₁)/(y₂ − y₁)Source: Midpoint formula and perpendicular bisector construction; standard analytic geometry, see OpenStax College Algebra Section 2.1.
Worked example
Two fire hydrants sit at (30, 80) and (150, 20) on a site plan in metres. Where should the equidistant standpipe go?
- Average the x values: (30 + 150) / 2 = 90.
- Average the y values: (80 + 20) / 2 = 50.
- Segment length: √(120² + 60²) = 134.16 m, so each hydrant is 67.08 m from the midpoint.
- Segment slope is −0.5, so the perpendicular bisector has slope 2.
Place it at (90, 50) — 67.08 m from each hydrant, and anywhere along the line of slope 2 through that point remains equidistant.
Frequently asked questions
Is the midpoint always on the segment?+
Yes, for a straight segment it lies exactly halfway along it. The formula never produces a point outside the span defined by the endpoints.
How do I find a point one third of the way along?+
Use a weighted split: x = x₁ + t(x₂ − x₁) with t = 1/3, and the same for y. The midpoint is that formula with t = 0.5.
What if the segment is vertical?+
The midpoint is still fine, but the perpendicular bisector slope becomes zero because the perpendicular is horizontal. A horizontal segment gives the reverse — an undefined perpendicular slope.
Does this find the centre of a shape?+
Only for a two-point segment. A polygon's centroid averages all vertices, and a weighted centre of mass needs each vertex's mass factored in.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.