Distance Between Two Points 2D Formula

This figure illustrates the formula for finding the distance between two points in two dimensions.

Formula

Summary

This formula calculates the distance between two points

Expression Description
The -coordinate of the first point.
The -coordinate of the first point.
The -coordinate of the second point.
The -coordinate of the second point.

Usage

To calculate the distance between two points in two dimensions the formula below can be used.

For example, to find the distance between the point and the point substitute their coordinates into the formula like so:

Calculate the result.

The distance between the two points and is units.

Examples

Example 1

Distance betweent he points (3,1) and (6,4)

This example demonstrates how to find the distance between the two points and .

Steps

  1. Start with the distance between two points (2D) formula.

  2. Substitute the coordinates of the points into the formula.

  3. Evaluate the subtraction expressions.

  4. Evaluate the absolute value expressions.

  5. Evaluate the exponent expressions.

  6. Evaluate the additon operation.

    Take the square root.

    The distance between the points and is equal to units.

Example 2

Distance betweent he points (2,5) and (7,3)

This example demonstrates how to find the distance between the two points and .

Steps

  1. Start with the distance between two points (2D) formula.

  2. Substitute the coordinates of the points into the formula.

  3. Evaluate the subtraction expressions.

  4. Evaluate the absolute value expressions.

  5. Evaluate the exponent expressions.

  6. Evaluate the addition expression.

    Optionally, take the square root.

    The distance between the points and is equal to units.

Example 3

Distance betweent he points (-1,3) and (2,-2)

This example demonstrates how to find the distance between the two points and .

Steps

  1. Start with the distance between two points (2D) formula.

  2. Substitute the coordinates of the points into the formula.

  3. Evaluate the subtraction expressions.

  4. Take the absolute value of the two expressions.

  5. Evaluate the exponent epxressions..

  6. Evaluate the addition operator.

    Optionally, take the square root.

    The distance between the points and is equal to units.

Explanation

The formula for the distance between the two points and can be derived using a combination of the Pythagorean Theorem and the distance between two points (1D) formula.

Derivation

  1. Observe that the geometry of the two points and forms the shape of a right triangle in the cartesian coordinate system. The hypotenuse labeled with the variable is equal to the distance between the two points. This is illustrated below.

    Distance betweent he points (-1,3) and (2,-2)

  2. Setup the equation for the pythagorean theorem.

  3. Rearrange the equation and take the square root of both sides.

  4. Find the lengths of adjacent and opposite of the right triangle by applying the one dimensional distance formula.

  5. Substitute these expressions into the expression from step three.

    Finally, change the variable to to represent distance and we have derived the formula.

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