As the name suggests, a distance formula expresses the distance between any two given points or the length of the line segment. In a two-dimensional plane, we use the Pythagoras theorem to derive the distance formula. It can be extended to find the distance between any two points in a three-dimensional plane as well.

**Distance Formula**

In coordinate geometry, there is a list of various distance formulas that can be applied to find out the distance between two points, the distance between a point and a line, the distance between any two parallel lines, the distance between two parallel planes, etc.

List of various types of distance formulas in coordinate geometry:

- The distance between any two given points in a 2D plane
- The distance between any two given points in a 3D plane
- The distance from a particular point to a line in 2D
- The distance between two lines that are parallel in 2D
- The distance from any point to a line in 3D
- The shortest distance between two skew lines
- The distance from a particular to a plane
- The distance between any two parallel planes

**Applications of Distance Formula**

The distance formula has several applications both in mathematics and in real-life situations.

- The distance between any given point and the origin can be found by using the distance formula.
- The formula that is used to find out the magnitude of a complex number is derived from the distance formula itself.
- To find out the distance between any two points in three-dimensional surfaces or n-dimensional planes, the distance formula is used.
- To find the magnitude of a vector by deriving the magnitude formula, the distance formula can be applied.
- If you want to find out the distance between any two points in a sea, then you have to identify the geographic coordinates and then apply the distance formula.
- If you travel from one city to another by air, then that shortest distance is also calculated by using the distance formula.
- To prepare maps and adjust the coordinates of new locations, surveyors and engineers take the help of the distance formula.
- Sailors use the distance formula as well. They apply it to determine their locations by using maps and coordinates in the sea.

**Formula of Distance Between Two Points**

**In a 2D plane**

The distance formula that is used to determine the distance between two points in a 2D plane is also known as the Euclidean distance formula. Let us assume two points in the 2D plane: A(x_{1},y_{1}) and B(x_{2},y_{2}), and the distance between A and B is d. By applying the Pythagoras theorem, we can derive the formula to calculate the distance. Therefore, the formula is,

d = √[(x_{2}-x_{1})^{2 }+ (y_{2}-y_{1})^{2}]

This is known as the distance between two points in a 2D plane formula.

**In a 3D plane**

To calculate the distance between the given two points in a 3D plane, let us assume two points: A(x_{1},y_{1},z_{1}) and B(x_{2},y_{2},z_{2}). By applying the Pythagoras theorem as it is used in the case of the 2D plane, we can determine the distance between the two points in a 3D plane,

d = √[(x_{2}-x_{1})^{2 }+ (y_{2}-y_{1})^{2} + (z_{2}-z_{1})^{2}]

**Applications of Distance Formula in Geometry**

Some of the applications of the distance formula in geometry are:

- If you have to prove that a given is a square shape, along with the four sides, as well as the diagonals, being equal, then the distance formula is used.
- Besides, to prove that a given figure is a rhombus and all four sides are equal, the distance formula is used.

To know more about the other types of distance formulas, visit Cuemath to learn some insights about this interesting subject.