Distance Formula
The distance formula gives the straight-line (Euclidean) distance between two points in the Cartesian plane using their coordinates. It is derived directly from the Pythagorean theorem by treating the horizontal and vertical separations as legs of a right triangle. The formula extends naturally to three dimensions as d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) and is fundamental in analytic geometry, physics, and data science.
Key Formula
d = sqrt((x2 - x1)² + (y2 - y1)²)
LaTeX: d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
| Symbol | Meaning | Unit |
|---|---|---|
| d | Distance between the two points | same as coordinate units |
| x₁, y₁ | Coordinates of the first point | unitless |
| x₂, y₂ | Coordinates of the second point | unitless |
Worked Example
Problem
Find the distance between points P(1, 2) and Q(7, 10).
Solution
Step 1 — Identify coordinates: x₁=1, y₁=2, x₂=7, y₂=10. Step 2 — Apply formula: d = √((7−1)² + (10−2)²) = √(6² + 8²) = √(36 + 64) = √100. Step 3 — Simplify: d = 10.
Answer
Distance PQ = 10 units
Distance Formula Applied to Common Point Pairs
| Point P | Point Q | Δx | Δy | Distance d |
|---|---|---|---|---|
| (0, 0) | (3, 4) | 3 | 4 | 5 |
| (1, 2) | (7, 10) | 6 | 8 | 10 |
| (−2, 1) | (2, 4) | 4 | 3 | 5 |
| (−5, −5) | (3, 1) | 8 | 6 | 10 |
| (0, 0) | (0, 7) | 0 | 7 | 7 (vertical) |
Interactive Tools
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Related Terms
Midpoint Formula
The midpoint formula gives the coordinates of the point that lies exactly halfway between two given points on a line segment in the Cartesian plane. It is derived by averaging the x-coordinates and the y-coordinates of the two endpoints separately. The midpoint is used extensively in geometry proofs, construction of perpendicular bisectors, and in applications such as GPS interpolation and computer graphics.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system in which points are represented by numerical coordinates. It provides a powerful bridge between algebra and geometry, allowing geometric shapes to be described by equations and algebraic operations to yield geometric insights. Key concepts include plotting points, finding distances, midpoints, slopes, and the equations of lines, circles, and conic sections on the Cartesian plane.
Pythagorean Theorem
The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (the legs). It is one of the most famous and widely applied theorems in mathematics, used in distance calculations, navigation, construction, and virtually every branch of science and engineering.
"Distance" comes from Latin "distantia" (a standing apart), from "distare" (to stand apart). The formula as applied to coordinates was a direct consequence of Descartes' analytic geometry (1637) combined with Pythagoras' theorem from ancient Greece (c. 570–495 BCE).