Euclidean Distance Formula
Before going to learn the Euclidean distance formula, let us see what is Euclidean distance. In coordinate geometry, Euclidean distance is the distance between two points. The two points are in a plane and the length of a segment connecting the two points is measured. It is the general way of representing the distance between two points. Let us learn the Euclidean distance formula along with a few solved examples.
What Is Euclidean Distance Formula?
We use the Pythagorean theorem to calculate the derive the Euclidean distance formula. Using this,
Distance between two points = √[(x\(_2\) – x\(_1\))^{2} + (y\(_2\) – y\(_1\))^{2}]
Where,
 (x\(_1\),y\(_1\)) are the coordinates of the first point.
 (x\(_2\),y\(_2\)) are the coordinates of the second point.
Let us see the applications of the Euclidean distance formula in the following section.

Example 1: Find the distance between points P(3, 2) and Q(4, 1).
Solution:
To find: Euclidean Distance
Given:
P(3, 2) = (x_{1}, y_{1})
Q(4, 1) = (x_{2}, y_{2})
Using Euclidean distance formula,
D = √[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]
PQ = √[(4 – 3)^{2} + (1 – 2)^{2}]
PQ = √[(1)^{2} + (1)^{2}]
PQ = √2 unit.
Answer: The Euclidean Distance between points A(3, 2) and B(4, 1) is √2 unit.

Example 2: Check that points A(√3, 1), B(0, 0), and C(2, 0) are the vertices of an equilateral triangle.
Solution:
To Find: The distance between AB, BC and, CA
Given:
A(√3, 1) =(x_{1}, y_{1})
B(0, 0)=(x_{2},y_{2})
C(2, 0)=(x_{3}, y_{3})
Using Euclidean distance formula,
AB = √[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]
BC = √[(x_{3} – x_{2})^{2} + (y_{3} – y_{2})^{2}]
CA = √[(x_{3} – x_{1})^{2} + (y_{3} – y_{1})^{2}]
AB = √[(0 – √3)^{2} + (01)^{2}] = √(3 + 1) = √4
BC = √[(20)^{2} + (00)^{2}] = √(4 + 0) = √4
CA = √[(2  √3)^{2}+ (0 – 1 )^{2}] = √(9 + 25) = 4(2√3)
Answer: AB, BC, and, CA are not the vertices of an equilateral triangle.