This definition is one way to say what almost all of us think of distance intuitively, but it is not the only way we could talk about distance. The most common meaning is the 1D space between two points. Referencing the above figure and using the Pythagorean Theorem,ĪC 2 = (x 2 - x 1) 2 + (y 2 - y 1) 2. We can visualize the 3D distance formula as a right triangle that happens to reside in the x-y-z 3D coordinate system. What is distance Before we get into how to calculate distances, we should probably clarify what a distance is. In a 3D coordinate plane, the distance between two points, A and B, with coordinates (x 1, y 1, z 1) and (x 2, y 2, z 2), can also be derived from the Pythagorean Theorem. Which is the distance formula between two points on a coordinate plane. We can rewrite this using the letter d to represent the distance between the two points as The horizontal and vertical distances between the two points form the two legs of the triangle and have lengths |x 2 - x 1| and |y 2 - y 1|. ![]() The hypotenuse of the right triangle, labeled c, is the distance between points A and B. Draw a line from the lower point parallel to the x-axis, and a line from the higher point parallel to the y-axis, then a right triangle will be formed. ![]() Given two points, A and B, with coordinates (x 1, y 1) and (x 2, y 2) respectively on a 2D coordinate plane, it is possible to connect the points with a line and draw vertical and horizontal extensions to form a right triangle: If you don’t want to memorize the formula, then there is another way to find the distance between the two points. Referencing the right triangle sides below, the Pythagorean theorem can be written as: The Pythagorean Theorem says that the square of the hypotenuse equals the sum of the squares of the two legs of a right triangle. The distance formula can be derived from the Pythagorean Theorem. ![]() The Pythagorean Theorem and the distance formula Other coordinate systems exist, but this article only discusses the distance between points in the 2D and 3D coordinate planes. Where (x 1, y 1, z 1) and (x 2, y 2, z 2) are the 3D coordinates of the two points involved. d =ĭistance formula for a 3D coordinate plane: Find the length of line segment AB given that points A and B are located at (3, -2) and (5, 4), respectively.
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