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

Common stuck point: The procedure for distance is the easy part; the trap is reporting a negative distance. Asking "Am I finding the shortest straight-line length between two specific points?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I finding the shortest straight-line length between two specific points?

Worked Examples

Example 1

easy

Find the distance between points

A(1, 2)

and

B(4, 6)

Answer

d = 5

units

First step

Step 1: Use the distance formula:

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

Full solution

2
Step 2: Substitute: $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2}$ .
3
Step 3: Calculate: $d = \sqrt{9 + 16} = \sqrt{25} = 5$ .

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.

Example 3

medium

Show that the points

(0,0), (5,0), (5, 5)

form a right isosceles triangle. Compute all side lengths.

Example 4

challenge

Prove that the distance function

d(A, B)

on the plane satisfies

d(A, B) = 0 \Leftrightarrow A = B

, symmetry, and the triangle inequality.

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.

Example 3

easy

Find the distance between

(0,0)

and

(3,4)

Example 4

easy

Find the distance between

(1,2)

and

(1,7)

Example 5

easy

Can a distance ever be negative?

Example 6

easy

Find the distance between

(2,3)

and

(2,3)

Example 7

easy

Find the distance between

(-2, 0)

and

(4, 0)

Example 8

easy

Why does the order of the two points not matter in the distance formula?

Example 9

easy

Find the distance between

(0,0)

and

(6,8)

Example 10

easy

What does 'distance' mean for two points: shortest path, or any path?

Example 11

medium

Find the distance between

(1, 2)

and

(4, 6)

Example 12

medium

How is straight-line (Euclidean) distance different from taxicab distance between

(0,0)

and

(3,4)

Example 13

medium

A point

(x, 0)

on the

x

-axis is equidistant from

(0, 3)

and

(6, 3)

. Find

x

Example 14

medium

Find the distance between

(-3, -1)

and

(1, 2)

Example 15

medium

Is triangle with vertices

(0,0)

(4,0)

(0,3)

a right triangle? Use distances.

Example 16

medium

A circle has center

(2, 1)

and passes through

(5, 5)

. Find its radius.

Example 17

medium

Find the perimeter of the triangle with vertices

(0,0)

(3,0)

(3,4)

Example 18

medium

Extend the distance formula to 3D: find the distance between

(0,0,0)

and

(2,3,6)

Example 19

challenge

Find the point on the

x

-axis equidistant from

A(1, 2)

and

B(5, 4)

Example 20

challenge

Three towns sit at

(0,0)

(8,0)

, and

(4, 3)

. Is the third town closer to the first or the second?

Example 21

challenge

Explain the triangle inequality

d(A,C) \leq d(A,B) + d(B,C)

in plain terms, and when equality holds.

Example 22

challenge

A point moves so it is always exactly

5

units from the origin. Using the distance formula, find the equation of its path.

Example 23

easy

Find the distance between

(0, 0)

and

(8, 15)

Example 24

easy

Find the distance between

(2, 5)

and

(2, -3)

Example 25

easy

Find the distance between

(-4, 1)

and

(3, 1)

Example 26

easy

Find the distance between

(7, 1)

and

(4, 5)

Example 27

easy

Find the distance between

(-2, -3)

and

(4, 5)

Example 28

medium

Find the distance between

(1, 1)

and

(5, 4)

Example 29

medium

Find the distance between

(3, 7)

and

(11, 22)

Example 30

medium

A circle is centered at

(1, 2)

and passes through

(4, 6)

. Find its radius.

Example 31

medium

Is the triangle with vertices

(0,0), (5,0), (0, 12)

a right triangle? Verify with side lengths.

Example 32

medium

Find the perimeter of the quadrilateral with vertices

(0,0), (4,0), (4,3), (0,3)

Example 33

medium

Find the distance between

(-1, 2)

and

(5, -6)

Example 34

medium

Find a point on the

y

-axis equidistant from

(2, 0)

and

(-3, 5)

Example 35

medium

Extend the distance formula to 3D: find the distance between

(1, 2, 3)

and

(4, 6, 15)

Example 36

hard

A point

(x, 0)

on the

x

-axis is

5

units from

(0, 4)

. Find all possible

x

Example 37

hard

Find the distance from the origin to the midpoint of the segment joining

(2, 6)

and

(8, 2)

Example 38

hard

Show that

(1, 2), (4, 6), (8, 3), (5, -1)

form a rhombus by computing all sides.

Example 39

hard

Find all points on the line

y = x

at distance

5

from the origin.

Example 40

hard

Find the distance between the parallel lines

y = 2x + 1

and

y = 2x - 3

Example 41

hard

A circle passes through

(0, 0), (6, 0), (0, 8)

. Find the center and radius using equidistance.

Example 42

hard

Let

A = (0,0)

and

B = (10, 0)

. Find the locus of points

P

such that

PA = 2 \cdot PB

Example 43

hard

Three points

A(0,0), B(4,0), C(2, h)

form an equilateral triangle. Find

h

Example 44

challenge

Find the minimum possible distance from a point on the line

x + y = 10

to the origin.

← Back to Distance Formula Explained Practice Mode

Related Concepts

Pythagorean Theorem Distance Formula

Background Knowledge

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

pythagorean theorem