Distance Formula

Last Updated : 10 Sep, 2026

The distance formula is used to find the distance between two points in cartesian coordinate system and can be calculated using the distance formula.

distance_formula

Let's say you have two points: A (x1​, y1​) and B (x2​, y2​).

Distance 'd' between A and B is given by the formula: d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}the

Derivation

Derivation of the Distance Formula is given using Pythagoras Theorem. In the right-angled triangle ABC, we have:

AB2 = AC2 + BC2

  • Distance between points A and C is (x2 - x1)
  • Distance between points B and C is (y2 - y1)

Distance, d is calculated as:

d^2 = {(x_2-x_1)^2 + (y_2-y_1)^2}

Now, taking the square root on both sides,

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

Thus, d is the distance between two points.

Distance between Two Points in Polar Co-ordinates

We can also calculate the distance between two points using the polar coordinates.

Let's take two points A (r1, θ1) and B (r2, θ2) then the distance between them is calculated using the Distance Formula:

AB = \sqrt{r_1^2 + r_2^2 - 2r_1 r_2 \cos(\theta_1 - \theta_2)}

Distance From a Point To a Plane

In 3D we can also calculate the distance between a point and the plane using the distance formula.

Let's take point A (x1, y1, z1) and plane P: ax + by + cz + d = 0 then the distance between the point and the plane is given using the Distance Formula:

d = \frac{|ax_1 + by_1 + cz_1|} { \sqrt{(a^2 + b^2 + c^2)}}

The image below shows the distance between the point and the plane:

5

Applications

Distance Formula has various applications in Mathematics, Sciences, and others and some of the most important applications of the Distance Formula are:

  • The distance of a point from the origin is calculated using the Distance Formula.
  • Distance between two points in 2D and 3D planes is calculated using the Distance Formula.
  • The magnitude of the vector is calculated using the Distance Formula.
  • The magnitude of the complex number is calculated using the Distance formula.

Apart from this distance formula also has some real-life applications which are:

  • Distance Formula is used to find the distance between two stars.
  • Distance Formula is also used to find the distance between various points in the sea and oceans.
  • Distance Formula is used to find the linear distance between two points in the Globe.

➢Practice: Solved Examples

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