Midpoint calculator with the midpoint formula and a plot

The midpoint of two points is halfway between them: average the x coordinates and average the y coordinates. So the midpoint of (1, 2) and (7, 5) is ((1 + 7)/2, (2 + 5)/2) = (4, 3.5). This midpoint calculator does that for any two points, gives the answer exactly as fractions, and also finds the distance between the points, the gradient (slope) and the equation of the line through them, all drawn on a coordinate plot.

It opens on (1, 2) and (7, 5). Type your own points, or open the full midpoint and distance calculator over this page.

Try it

Midpoint and Distance Calculator

Find the midpoint, distance and gradient between two points

Open the calculatorGo to its page

How to use the midpoint calculator

1. Type the first point in A: x and A: y, and the second in B: x and B: y. Negative numbers and decimals are fine.
2. Or drag A and B on the coordinate plot. They snap to the grid, and every answer updates as you move.
3. Read the Midpoint at the top, with the Distance, the Gradient and the Line through both points beside it.
4. The working below shows each formula with your numbers in it. The plot marks the midpoint and draws the right-angled triangle used for the distance.
5. To practise, press Random problem in the Practice section.

Answers are exact: a midpoint like (4, 3.5) shows as (4, 7/2), and a distance like √45 shows as 3√5 with the decimal 6.708204 beside it. Coordinates can go up to a million either way.

The midpoint formula

Each coordinate of the midpoint is the mean of the two matching coordinates. It doesn't matter which point you call the first: adding is the same either way round. The midpoint splits the segment in the ratio 1 : 1, the simplest case of dividing a segment in a given ratio, which the Common Core standards for high school geometry cover.

Worked examples you can check in the calculator

Example 1: the midpoint of a line segment

Find the midpoint of A(1, 2) and B(7, 5), the calculator's opening points.

The midpoint is (4, 3.5). The calculator also gives the distance, √(6² + 3²) = √45 = 3√5 ≈ 6.708, the gradient, 3/6 = 1/2, and the line y = ½x + 3/2.

Example 2: negative coordinates

Find the midpoint of (−3, 4) and (5, −2). Take care with the signs as you add:

The midpoint is (1, 1). The points are 8 apart across and 6 apart up, so the distance between them is exactly 10.

Example 3: decimals

Find the midpoint of (2.5, −1) and (−4, 6). The x coordinate is (2.5 + (−4)) ÷ 2 = −1.5 ÷ 2 = −0.75, and the y coordinate is (−1 + 6) ÷ 2 = 2.5. The midpoint is (−0.75, 2.5), which the calculator writes as (−3/4, 5/2).

Example 4: find the endpoint given the midpoint

M(3, −2) is the midpoint of A(−1, 4) and B. Find B. The midpoint is halfway, so B is as far past M as A is before it. Double the midpoint and take away A:

So B = (7, −8). The calculator doesn't work backwards, but you can check: type A(−1, 4) and B(7, −8), and the midpoint comes out as (3, −2).

Example 5: the perpendicular bisector

The perpendicular bisector of AB passes through its midpoint at right angles to it. For A(2, 1) and B(6, 9), the calculator gives the midpoint (4, 5) and the gradient 8/4 = 2. A perpendicular line has gradient −1/2, the negative reciprocal. Through (4, 5):

Every point on this line is the same distance from A as from B. It is a common exam question because it uses the midpoint and the gradient together.

Try one

Distance, gradient and the line

The calculator gives three more answers for the same two points, since questions often ask for them together:

The distance formula is the Pythagorean theorem: the run and the rise are the two shorter sides of a right triangle, and the distance is the hypotenuse. The Pythagorean theorem calculator guide explains it. When the two x values are equal the line is vertical, and the calculator says the gradient is undefined.

Common mistakes

• Subtracting instead of adding. The midpoint adds the coordinates. Subtracting is for the distance and the gradient.
• Forgetting to halve. (x₁ + x₂) on its own is twice the midpoint's x.
• Mixing x and y. Average the x values with each other and the y values with each other.
• Sign slips with negatives. 4 + (−2) = 2. Write negatives in brackets as you substitute.
• Halving the endpoint instead of doubling the midpoint when you are finding a missing endpoint. B = 2M − A.

Read next

Maths · GeometryLine segment: definition, notation, length and midpointWhat a line segment is, how it differs from a line and a ray, how to name it, and how to find its length and midpoint on a number line or a graph.