The distance between two points

The distance between two points in the coordinate plane comes from the Pythagorean theorem¹. Here we find the distance between $A(1, 2)$ and $B(4, 6)$ .

Building a right triangle

Going from $A$ to $B$ moves $4 - 1 = 3$ across and $6 - 2 = 4$ up. Taking these horizontal and vertical differences as the two legs of a right triangle, the segment joining $A$ and $B$ is its hypotenuse.

AB = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

The familiar right triangle with sides in the ratio $3 : 4 : 5$ has appeared.

The general formula

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Because the differences are squared, it does not matter if $x_2 - x_1$ is negative: naming the points in either order gives the same distance.

Situation	Formula
General pair of points	$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Distance from the origin	$\sqrt{x^2 + y^2}$
On the number line	$\|x_2 - x_1\|$

The distance from the origin is what the absolute value on the number line becomes in the plane.

The midpoint

The same two points also give the midpoint of the segment, whose coordinates are the averages of the $x$ values and of the $y$ values.

\left(\frac{1 + 4}{2}, \frac{2 + 6}{2}\right) = \left(\frac{5}{2}, 4\right)

The equation of a circle

A circle is the set of points at distance $r$ from a center $(a, b)$ , so $\sqrt{(x - a)^2 + (y - b)^2} = r$ , and squaring gives the familiar equation.

(x - a)^2 + (y - b)^2 = r^2

The equation of a circle is nothing but the distance formula rewritten.

Identifying a triangle

Identifying the shape of a triangle from three points also begins here, by computing its three side lengths.

Condition on the sides	The triangle is
two sides equal	isosceles
$a^2 + b^2 = c^2$ for the longest side $c$	right-angled
all three sides equal	equilateral

The distance is $0$ only when the two points coincide; otherwise it is always positive.

Extending to space

In three dimensions you add the square of the difference in $z$ . This is the Euclidean distance², and it remains the basic tool in applications such as nearest-neighbor search, where the closest data point is the one to find.

The large dots on the graph are $A$ and $B$ , and the length of the segment joining them is the distance $5$ we found.

Pythagorean theorem, Wikipedia
Euclidean distance, Wikipedia