Distance between two points
d = √[(x₂−x₁)² + (y₂−y₁)²] — Pythagoras on the coordinate grid.
Key formulas
![d = √[(x₂−x₁)²+(y₂−y₁)²]](/formulas/01ae67e9e50cf196.png "d = √[(x₂−x₁)²+(y₂−y₁)²]")
Origin story
The horizontal run is x₂−x₁ and the vertical rise is y₂−y₁. Those legs form a right triangle whose hypotenuse is the distance.
Example
A(1,2), B(4,6) → d=√(9+16)=5.
Order
Swapping the points only changes signs inside the squares, so d is unchanged.
Worked example
Worked example
Problem. Find the distance between A(1, 2) and B(4, 6).
Solution. d = √[(4−1)² + (6−2)²] = √(9+16) = √25 = 5.
3-4-5
Differences 3 and 4 always give distance 5.
Review quiz
Distance (−2,3) to (3,−9) is:
Practice questions
- Find distance from (−2,3) to (4,−5).
- Show that (0,0), (6,0), (0,8) form a right triangle by distances.
Study notes
Do I need absolute values?
Squaring removes the sign; absolute value is optional beforehand.