Distance Between Two Points
Simple Explanation
The distance between two points is the straight-line length between them. It is found by treating the horizontal and vertical gaps between the points as the two shorter sides of a right triangle, and applying the Pythagorean theorem.
Why Do We Need It?
Distance lets you measure length and separation directly from coordinates — the basis for geometry, navigation, and physics calculations.
See It
A coordinate plane with two draggable points and the line segment between them, showing the live distance value
Formula
Distance Formula
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
The straight-line distance between two points on a coordinate plane.
- d
- — the distance between the two points
- x₁, y₁
- — the coordinates of the first point
- x₂, y₂
- — the coordinates of the second point
When to use it: Whenever you know the coordinates of two points and need the straight-line distance between them.
Worked Example
Find the distance between two points
Find the distance between A(1, 2) and B(4, 6).
Why Does This Work?
The formula is the Pythagorean theorem applied to the horizontal and vertical distances between the points, which form a right triangle whose hypotenuse is the segment AB.
Real-Life Example
Straight-line distance on a map grid
A city-planning map places two buildings at known grid coordinates.
The distance formula instantly gives the straight-line ("as the crow flies") distance between them, without needing to measure by hand.
Practice
Find the distance between (0, 0) and (3, 4).
EasyFind the distance between (-2, 1) and (3, 13).
MediumCommon mistake
Forgetting to square the differences before adding them (writing d = (x₂−x₁) + (y₂−y₁) instead of using squares and a square root).
Quick Review
- d = √[(x₂−x₁)² + (y₂−y₁)²]
- It comes directly from the Pythagorean theorem.
- The order of the two points does not matter — squaring removes the sign.