Distance Formula Calculator

The distance formula computes the straight-line distance between two points in coordinate space. It's a direct consequence of the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical separation between the points.

2D form — for points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

3D form — for points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

$n$-dimensional form (Euclidean distance):

$d = \sqrt{\sum_{i=1}^n (b_i - a_i)^2}$

This generalizes naturally to any number of dimensions, which is why it's the workhorse 'distance' notion in physics, statistics, and machine learning.

Step-by-Step

1. Label the points $(x_1, y_1)$ and $(x_2, y_2)$. Either assignment works — the formula is symmetric.
2. Compute differences: $\Delta x = x_2 - x_1$, $\Delta y = y_2 - y_1$.
3. Square them: $(\Delta x)^2$ and $(\Delta y)^2$.
4. Sum: $(\Delta x)^2 + (\Delta y)^2$.
5. Take the square root: $d = \sqrt{\text{sum}}$.
6. Simplify the radical if possible (e.g., $\sqrt{50} = 5\sqrt{2}$).

Geometric Derivation

Draw a horizontal segment from $(x_1, y_1)$ to $(x_2, y_1)$ — length $|x_2 - x_1|$.
Draw a vertical segment from $(x_2, y_1)$ to $(x_2, y_2)$ — length $|y_2 - y_1|$.
The original segment is the hypotenuse of a right triangle with these two legs, so by the Pythagorean theorem:

$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$

Taking square roots gives the distance formula. The absolute values aren't needed because squaring removes the sign.

Related Formulas

• Midpoint: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ — the average of the coordinates.
• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$ — uses the same differences as the distance formula.
• Distance from point to origin: $d = \sqrt{x^2 + y^2}$ (special case with $(x_1, y_1) = (0, 0)$).

Manhattan / Taxicab Distance (For Comparison)

Note that the formula above is Euclidean distance. Manhattan distance $|x_2 - x_1| + |y_2 - y_1|$ measures travel on a grid (no diagonals). They are different metrics — make sure you know which one your problem wants.

• Forgetting to square: $d \ne (x_2 - x_1) + (y_2 - y_1)$. The squares (and the square root) are essential.
• Sign errors: $(x_2 - x_1)^2 = (x_1 - x_2)^2$, so subtraction order doesn't matter — but only because of the square. Don't drop the square because you 'see' the difference.
• Forgetting to take the square root: $(x_2 - x_1)^2 + (y_2 - y_1)^2$ is $d^2$, not $d$. Many students stop one step early.
• Not simplifying the radical: $\sqrt{8} = 2\sqrt{2}$. Leaving as $\sqrt{8}$ is technically correct but usually marked down on exams.
• Mixing 2D and 3D: If your problem is in 3D, include the $(z_2 - z_1)^2$ term. If 2D, don't invent a $z$ term.