Distance Calculator (Two Points)

1. Enter the x and y of the first point.
2. Enter the x and y of the second point.
3. Read the distance, the straight line between them.
4. See the horizontal and vertical change that make up that line.

Here is what each result means:

The distance between two points is the length of the straight line joining them. On a coordinate grid, it is found from the horizontal and vertical gaps between the points using the distance formula. From (0, 0) to (3, 4), the distance is 5.

This is really the Pythagorean theorem in disguise. The horizontal and vertical changes are the two legs of a right triangle, and the distance is its hypotenuse. That link makes the distance formula easy to remember and applies to maps, physics and computer graphics alike.

The result is always positive or zero, because a length cannot be negative. It does not matter which point you call first, and swapping them leaves the distance unchanged. What matters is the size of the gap in each direction, since those two gaps set the length of the line between the points.

It squares the horizontal and vertical changes, adds them, and takes the square root.

1. Subtract the x values for the horizontal change, and the y values for the vertical change.
2. Square each change and add them.
3. Take the square root of the total for the distance.

Because the two changes form a right triangle, the Pythagorean theorem calculator gives the same length from the legs.

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

Calculation: the horizontal change is 3 and the vertical change is 4, so d = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. The two points are exactly 5 units apart, the classic 3-4-5 right triangle.

The straight-line distance is always at least as long as either single change, and never longer than their sum.

When one change is zero, the distance equals the other change, since the line lies straight along an axis.

The distance formula and the Pythagorean theorem are the same idea in two guises.

So if you can use the Pythagorean theorem, you already know the distance formula.

The Gaps in Each Direction

Larger horizontal or vertical gaps make a longer line. Both count toward the total through their squares.

Diagonal Versus Straight

A diagonal path is longer than either axis gap alone but shorter than walking along both, the triangle inequality.

The Order of the Points

Swapping the two points does not change the distance, since squaring removes the sign of each change.

Geometry Homework

Find the length of a segment between two coordinates.

Maps and Design

Measure straight-line gaps on a grid, plan or screen layout.

Physics and Graphics

Compute how far apart two positions are for motion or collision checks.

1. Forgetting to Square

Each change must be squared before adding. Adding the raw changes gives the wrong length.

2. Skipping the Square Root

The sum of squares is the distance squared. Take the square root for the actual distance.

3. Sign Slips with Negatives

Subtracting a negative coordinate adds. Track the signs, though squaring fixes the final sign.

4. Mixing Up the Coordinates

Pair x with x and y with y. Crossing them changes the answer.

5. Using It on a Globe

The formula assumes a flat plane. Long distances on Earth need a great-circle method instead.

The distance formula is exact; only the displayed decimals are rounded.

Subtract the x values and the y values, square each result, add them, and take the square root. From (0, 0) to (3, 4), that is the square root of 9 plus 16, which is 5.

It is d equals the square root of the squared horizontal change plus the squared vertical change. It comes straight from the Pythagorean theorem applied to the two coordinate gaps.

Yes, in effect. The horizontal and vertical changes are the two legs of a right triangle, and the distance is the hypotenuse, so the distance formula is the theorem written for coordinates.

No. Swapping the two points gives the same distance, because each change is squared, which removes its sign. Distance is always positive or zero.

The distance is zero. Both changes are zero, so the square root of zero is zero. The tool reports this case directly.

Only for short distances where the ground is nearly flat. Long distances on the globe follow a curved great-circle path and need a different formula.

Distance here is the straight-line length between two points and is always positive. Displacement in physics also has a direction, though its size equals this distance.

This tool works in two dimensions. In three dimensions you add the squared change in z under the same square root before taking the root.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Distance (Wikipedia).
• The distance between two points (Maths Is Fun).
• Pythagorean theorem (Wikipedia).