Distance Formula

Geometry · Axiom Academy

Calculate the length between any two points on a coordinate plane — straight from the Pythagorean theorem. Take any two points (x_1, y_1) and (x_2, y_2) . The horizontal gap between them is and the vertical gap is . Those gaps are the two legs of a right triangle, and the distance d between the points is its hypotenuse . solve for the hypotenuse — that is the distance formula Plot A(1, 1) and B(8, 5) . Walking across gives a horizontal leg of ; walking up gives a vertical leg of . The straight segment from A to B closes the triangle as its hypotenuse. — the run from A across to below B . — already simplified ( 65 has no square factors). Find the distance between A(-2, 3) and B(4, -1) . The same three moves work even when coordinates are negative — just track the signs carefully. hides a perfect-square factor: , so . Always pull out square factors. The negative squares to +16 , so a wrong sign on a difference can never break the answer. You can now find the distance between any two points by treating it as the hypotenuse of a right triangle — the Pythagorean theorem, written in coordinates. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Geometry course.