Distance Calculator

What this calculator does

Two points on a grid, and you want the straight line between them, the as-the-crow-flies gap rather than the path around. That is the distance formula, and here is the nice part: it is really the Pythagorean theorem in disguise. The horizontal gap and the vertical gap are the two legs of a right triangle, and the distance is the hypotenuse.

Type the coordinates of both points, and the tool measures the gap.

Using the calculator

1. Enter the x and y of the first point.
2. Enter the x and y of the second point.
3. Press Calculate.

The result is the distance, given exactly and rounded.

The distance formula

The distance between two points comes from the distance formula:

d = √((x2 − x1)² + (y2 − y1)²)

You find the horizontal gap between the points, then the vertical gap, square each of them, add the two, and take the square root of the total.

It is really Pythagoras

The formula may look new, but it is the Pythagorean theorem in disguise. The two points, with the horizontal and vertical gaps between them, form a right triangle. Those gaps are the two legs, and the straight-line distance is the hypotenuse. So d = √(leg² + leg²) is simply a² + b² = c² rearranged to give the long side.

A worked example | (0, 0) to (3, 4)

Find the distance from (0, 0) to (3, 4).

1. Horizontal gap 3, vertical gap 4.
2. d = √(3² + 4²) = √(9 + 16) = √25 = 5.

The familiar 3-4-5 right triangle, one more time.

What it is for

Distance on a coordinate plane turns up throughout geometry and graphing, in mapping and navigation, and anywhere you need the length of a segment or the gap between two locations. It is one of the first things you reach for when working with points on a grid.

Questions people ask

What is the distance formula?

d = √((x2 − x1)² + (y2 − y1)²), the square root of the squared gaps added together.

How does it relate to Pythagoras?

The horizontal and vertical gaps are the legs of a right triangle, and the distance is the hypotenuse.

What is the distance from (0, 0) to (3, 4)?

5, since √(9 + 16) = √25 = 5.

Does it give an exact answer?

Yes, it shows both the exact distance and a rounded value.

References

A note on the idea behind it. The distance between two points is d = √((x2 − x1)² + (y2 − y1)²), which is the Pythagorean theorem applied to the horizontal and vertical gaps between the points. For further reading, see Euclidean distance.

1. The distance formula for two points, the square root of the sum of the squared coordinate differences.