Midpoint Calculator
With x₁ 2, y₁ 3, x₂ 8, y₂ 11, midpoint comes to 10.0000 — distance. It is reached in 5 steps, the last of which is sqrt(6 * 6 + 8 * 8), and each one is printed on the page with its numbers filled in. The formula is the one published by NIST Digital Library of Mathematical Functions, not an approximation fitted to it.
The midpoint of two points, with the distance, the slope and the perpendicular bisector — what a coordinate-geometry question asks about the same pair.
Formula and sources checked · How we check
x₁ 2, y₁ 3, x₂ 8, y₂ 11
10.0000
Distance for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Midpoint x
(2 + 8) / 25- Midpoint y
(3 + 11) / 27- Distance between the points
sqrt(6 * 6 + 8 * 8)10- Slope of the line through them
8 / 61.333- Slope of the perpendicular bisector
-6 / 8-0.75
Worked example
The midpoint of (2, 3) and (8, 11) is (5, 7) — the average of the x values and the average of the y values. The two points are 10 apart, because 6 and 8 are the legs of a 6-8-10 right triangle.
How to work it out yourself
- 1.Average the x coordinates, then average the y coordinates. That is the entire midpoint formula, and writing it as (x₁ + x₂)/2 makes it look harder than "halfway between".
- 2.The distance uses Pythagoras on the same two differences, which is why the two formulas are always taught together.
- 3.The perpendicular bisector passes through the midpoint with the negative reciprocal slope. A line of slope 4/3 has a bisector of slope −3/4.
- 4.For three dimensions, average the z coordinates too and add z² under the square root. Nothing else changes.
Distance from the origin, second point varied along y
| y₂ | Distance | Midpoint y | Slope of the line through them |
|---|---|---|---|
| 0 | 6.7082 | 1.5 | -0.5 |
| 1 | 6.3246 | 2 | -0.33 |
| 2 | 6.0828 | 2.5 | -0.17 |
| 3 | 6.0000 | 3 | 0 |
| 4 | 6.0828 | 3.5 | 0.17 |
| 5 | 6.3246 | 4 | 0.33 |
| 6 | 6.7082 | 4.5 | 0.5 |
| 8 | 7.8102 | 5.5 | 0.83 |
| 10 | 9.2195 | 6.5 | 1.17 |
| 12 | 10.8167 | 7.5 | 1.5 |
| 15 | 13.4164 | 9 | 2 |
The midpoint moves half as fast as the endpoint, which is the whole of the formula: it is the average of the coordinates, not a construction.
The formula
- Midpoint x
(2 + 8) / 2 - Midpoint y
(3 + 11) / 2 - Distance between the points
sqrt(6 * 6 + 8 * 8) - Slope of the line through them
8 / 6 - Slope of the perpendicular bisector
-6 / 8
Source: NIST Digital Library of Mathematical Functions — coordinate geometry, Euclid, Elements Book I, Proposition 10 — bisecting a given line segment
Questions people actually ask
- What is the midpoint formula?
- ((x₁ + x₂)/2, (y₁ + y₂)/2). It is the average of the two points, taken one coordinate at a time. The midpoint of (2, 3) and (8, 11) is (5, 7).
- How do you find the endpoint if you know the midpoint?
- Double the midpoint and subtract the endpoint you have: x₂ = 2m − x₁. If the midpoint is (5, 7) and one end is (2, 3), the other is (8, 11).
- What is a perpendicular bisector?
- The line at right angles to a segment through its midpoint. Every point on it is equidistant from the two endpoints, which is why the circumcentre of a triangle is where the three perpendicular bisectors meet.
- Is the midpoint the same as the average?
- Yes — that is exactly what it is, done separately for each coordinate. The word "midpoint" describes the geometry and the word "average" describes the arithmetic, and it is the same operation.
- Does this work in three dimensions?
- The same way: average each coordinate, and the distance becomes √(Δx² + Δy² + Δz²). The formula generalises to any number of dimensions without changing shape.
Related
- Distance Formula CalculatorDistance between two points in the plane or in space, with the midpoint and the slope, and every step of the square root shown.
- Slope CalculatorSlope, intercept, angle and the full line equation from two points, with the perpendicular slope beside it.
- Dot and Cross ProductDot product, cross product, magnitudes and the angle between two 3-D vectors. The dot returns a number, the cross returns a vector — that is the difficulty.
- Interpolation CalculatorThe value between two known points on a straight line, with the slope it assumes and a warning when the target sits outside them.
- Diagonal & Rafter Length CalculatorHypotenuse of a right triangle in feet and inches — brace lengths, rafter runs, and squaring a layout.
- Circle CalculatorArea, circumference and radius of a circle from its diameter — in square feet and square inches.
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