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 straight-line distance between two points is the hypotenuse of the right triangle made by their coordinate differences.

Common stuck point: The procedure for distance formula is the easy part; the trap is forgetting to square the differences before adding. Asking "Do I have two points' coordinates and need the length of the segment joining them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have two points' coordinates and need the length of the segment joining them?

Worked Examples

Example 1

easy

Find the distance between the points

(1, 2)

and

(4, 6)

Find the distance d between A and B.

Answer

d = 5

First step

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}

Full solution

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
Substitute: $d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ . Recognise the 3-4-5 Pythagorean triple — no calculator needed.

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)

Find the distance d between A and B.

Example 3

medium

Show that the points

A(1, 1), B(4, 5), C(8, 2)

form a right triangle.

Verify triangle ABC is a right isosceles triangle.

Example 4

challenge

Derive the distance formula from the Pythagorean theorem starting with two general points

P_1(x_1, y_1)

and

P_2(x_2, y_2)

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.

Show triangle ABC is isosceles.

Example 2

medium

Find the points on the

x

-axis that are

5

units away from the point

(2, 4)

Example 3

easy

Find the distance between

(2, 3)

and

(7, 15)

Find the distance between A(2,3) and B(7,15).

Example 4

easy

Find the distance between

(-1, 4)

and

(2, 0)

Find the distance between A(-1,4) and B(2,0).

Example 5

easy

Find the distance between

(6, -2)

and

(6, 7)

Find the distance between A and B (same x-coordinate).

Example 6

easy

Find the distance between

(0, 0)

and

(9, 12)

Find the distance from the origin to P(9,12).

Example 7

easy

Find the distance between

(3, 8)

and

(0, 4)

Find the distance between A(3,8) and B(0,4).

Example 8

medium

Find the distance from

(2, 3)

(2 + 6\sqrt{2}, 3 + 6\sqrt{2})

Example 9

medium

Find the distance between

(-5, 2)

and

(3, -4)

, in simplest radical form.

Find the distance between A(-5,2) and B(3,-4).

Example 10

medium

A circle has equation

(x - 4)^2 + (y + 1)^2 = 25

. Is the point

(8, 2)

on, inside, or outside the circle?

Example 11

medium

Find the perimeter of the triangle with vertices

(0, 0), (8, 0), (4, 6)

Find the perimeter of triangle ABC.

Example 12

medium

Find the distance between

(1, -2)

and

(4, 2)

Find the distance between A(1,-2) and B(4,2).

Example 13

medium

A square has vertices

(0,0), (a, 0), (a, a), (0, a)

. Find the length of its diagonal in terms of

a

Example 14

medium

Find a point on the

y

-axis at distance

\sqrt{50}

from

(5, 0)

Example 15

medium

Find the 3D distance between

(0, 0, 0)

and

(3, 4, 12)

Example 16

hard

The midpoint of segment

AB

(3, 5)

and

A = (-1, 2)

. Find

B

and the length of

AB

Midpoint M=(3,5), A=(-1,2). Find B and |AB|.

Example 17

hard

Determine whether the points

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

form a rectangle by computing the diagonals.

Both diagonals equal 5 — confirm this is a rectangle.

Example 18

hard

Find the point on the line

y = 2x

closest to the point

(5, 0)

Example 19

hard

(x, y)

is equidistant from

(0, 0)

and

(6, 8)

, find the equation relating

x

and

y

Example 20

hard

Find the distance between the centers of two circles

x^2 + y^2 = 4

and

(x - 7)^2 + (y - 24)^2 = 9

, and determine whether they intersect.

Example 21

hard

A regular hexagon has center at the origin and a vertex at

(4, 0)

. Find the distance between two adjacent vertices.

Example 22

hard

Three towns at

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

. Which two are farthest apart?

Which two towns are farthest apart?

Example 23

hard

Find all

a

such that

(a, 2a)

is at distance

\sqrt{45}

from the origin.

Example 24

challenge

Find the area of the triangle with vertices

A(0, 0), B(6, 0), C(2, 4)

using the distance formula plus Heron's formula.

Find the area using the distance formula and Heron's formula.

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Related Concepts

Pythagorean Theorem Coordinate Plane Square Roots Midpoint Formula Coordinate Proofs Equation of a Circle

Background Knowledge

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

pythagorean theoremcoordinate planesquare roots