Distance Examples in Math

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

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

Concept Recap

The length of the shortest path between two points, always a non-negative real number.

'As the crow flies'—the straight-line separation between two locations.

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: Distance measures separation between two points; it is always non-negative and zero only when the points coincide.

Common stuck point: Distance on a plane uses the Pythagorean theorem (distance formula).

Sense of Study hint: Draw a right triangle between the two points. The horizontal and vertical legs give you the values to plug into the Pythagorean theorem.

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: Use the distance formula: d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.
  2. 2
    Step 2: Substitute: d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2}.
  3. 3
    Step 3: Calculate: d = \sqrt{9 + 16} = \sqrt{25} = 5.

Answer

d = 5 units
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. Here the 3-4-5 right triangle makes the answer a whole number.

Example 2

medium
Find the distance between P(-2, 3) and Q(4, -1). Leave your answer in simplest radical form.

Practice Problems

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

Example 1

easy
What is the distance from the origin (0,0) to the point (5, 12)?

Example 2

hard
Find the distance between points A(1, 2, 3) and B(4, 6, 3) in 3D space.
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Background Knowledge

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

pythagorean theorem