Distance Formula Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Distance Formula.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A formula for finding the distance between two points in the coordinate plane, derived directly from the Pythagorean theorem.

Imagine two points on a grid. Draw a horizontal line from one and a vertical line from the other to form a right triangle. The horizontal leg is the difference in x-coordinates, the vertical leg is the difference in y-coordinates, and the hypotenuse—the direct distance—comes from the Pythagorean theorem. The distance formula is just a^2 + b^2 = c^2 in coordinate clothing.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The distance formula works because any two points on a coordinate plane form the hypotenuse of a right triangle whose legs are the horizontal and vertical distances. Applying the Pythagorean theorem (a^2 + b^2 = c^2) to those legs gives the direct distance.

Common stuck point: The order of subtraction doesn't matter ((x_2 - x_1)^2 = (x_1 - x_2)^2) because squaring eliminates the sign.

Worked Examples

Example 1

easy
Find the distance between the points (1, 2) and (4, 6).

Solution

  1. 1
    The distance formula is derived from the Pythagorean theorem: the horizontal and vertical separations form the legs of a right triangle, and the distance is the hypotenuse. d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
  2. 2
    Identify the coordinates: (x_1, y_1) = (1, 2) and (x_2, y_2) = (4, 6). Compute the differences: x_2 - x_1 = 3, y_2 - y_1 = 4.
  3. 3
    Substitute: d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. Recognise the 3-4-5 Pythagorean triple — no calculator needed.

Answer

d = 5
The distance formula is a direct application of the Pythagorean theorem on the coordinate plane. The horizontal and vertical differences form the legs of a right triangle, and the distance is the hypotenuse.

Example 2

medium
Find the distance between (-3, 5) and (2, -7).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Show that the triangle with vertices A(0, 0), B(3, 4), and C(6, 0) is isosceles.

Example 2

medium
Find the points on the x-axis that are 5 units away from the point (2, 4).
← Back to Distance Formula Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

pythagorean theoremcoordinate planesquare roots