Midpoint and distance calculator, with the steps.

The midpoint formula finds the point exactly halfway between two points: add the two x values and halve them, then do the same with the y values. The distance formula gives the length of the segment between them, using Pythagoras. This calculator does both for any two points, gives the distance as an exact root and as a decimal, and shows each step with your numbers put in. It also finds the slope, the perpendicular bisector, the midpoint and distance in three dimensions, a missing endpoint, and the point that splits a segment in a ratio.

How to use the calculator

1. Under Two points, type the x and y of Point 1 and Point 2. The answer updates as you type.
2. Read the cards: the midpoint, the distance, the slope, half the distance and the perpendicular bisector. Under them is The working, and below that the graph, with the right triangle drawn dotted.
3. Choose 3D points to add a z value for each point.
4. Choose Find an endpoint when you know one end and the midpoint. Type them in to get the other end.
5. Choose Split in a ratio, type the two parts, or click a button like 1:2 a third, to find a point along the segment.
6. Swap the points flips the two points, Copy result copies the answer as a line, and Copy link copies a link with your numbers in it.

The formulas

Worked examples

• Two points, (2, 3) and (6, 11). The midpoint is ((2 + 6)/2, (3 + 11)/2) = (4, 7). The gaps are 6 − 2 = 4 and 11 − 3 = 8, so d = √(16 + 64) = √80. Since 80 = 16 × 5, that is 4√5, about 8.9443. The slope is 8/4 = 2.
• Negative numbers, (−3, 4) and (5, −2). The midpoint is ((−3 + 5)/2, (4 + (−2))/2) = (1, 1). The gaps are 5 − (−3) = 8 and −2 − 4 = −6, so d = √(64 + 36) = √100 = 10. Take care: taking away a negative number adds.
• Fractions, (1/2, 1/3) and (3/2, 5/3). The midpoint is (1, 1). The gaps are 1 and 4/3, so d² = 1 + 16/9 = 25/9 and d = 5/3.
• Three dimensions, (1, 2, 3) and (4, 6, 15). The midpoint is (5/2, 4, 9). The gaps are 3, 4 and 12, so d = √(9 + 16 + 144) = √169 = 13.
• A missing endpoint. One end is (2, 3) and the midpoint is (4, 7). The other end is x = 2 × 4 − 2 = 6 and y = 2 × 7 − 3 = 11, which is (6, 11). Check: the midpoint of (2, 3) and (6, 11) is (4, 7).
• A ratio. To find the point one third of the way from (0, 0) to (9, 6), use m : n = 1 : 2. The x value is (2 × 0 + 1 × 9)/3 = 3 and the y value is (2 × 0 + 1 × 6)/3 = 2, so the point is (3, 2). The whole segment is √117 = 3√13, and the point is √13, about 3.6056, from the start.

The perpendicular bisector

The perpendicular bisector is the line that crosses the segment at its midpoint at a right angle. Its slope is the negative reciprocal of the segment's slope. For (2, 3) and (6, 11) the slope is 2, so the bisector has slope −1/2. It goes through (4, 7), so b = 7 − (−1/2)(4) = 9 and the line is y = −x/2 + 9. If the segment is flat, the bisector is a vertical line x = the midpoint's x. If the segment is vertical, the bisector is a flat line.

Common problems

• Which point is first. It does not matter. Swapping the points gives the same midpoint and the same distance, because the gaps are squared.
• Forgetting to halve. The midpoint is the sum divided by 2, not the sum.
• Leaving the root. √80 is a fine exact answer, and 4√5 is the tidy form. The decimal is rounded.
• Same point twice. The midpoint is that point and the distance is 0, so there is no bisector.

For the equation of the whole line, with both intercepts, use the slope calculator. For a right triangle with a known side, try the triangle calculator, and to work with fractions use the fraction calculator.