Pythagorean Theorem Examples in Math

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

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

Concept Recap

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

If you draw squares on each side of a right triangle, the two smaller squares fill the big one exactly.

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 theorem connects the side lengths of right triangles only.

Common stuck point: The procedure for pythagorean theorem is the easy part; the trap is using the theorem without a right angle. Asking "Do I know which side is the hypotenuse?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I know which side is the hypotenuse?

Worked Examples

Example 1

easy

A right triangle has legs of length

3

and

4

. Find the hypotenuse.

Answer

c = 5

First step

Apply the Pythagorean theorem:

a^2 + b^2 = c^2

Full solution

2
Substitute: $3^2 + 4^2 = c^2 \Rightarrow 9 + 16 = c^2 \Rightarrow c^2 = 25$ .
3
Take the positive square root: $c = \sqrt{25} = 5$ .

The Pythagorean theorem relates the three sides of a right triangle. The hypotenuse (opposite the right angle) is always the longest side. The

3

4

5

triple is the most common Pythagorean triple.

Example 2

medium

A right triangle has a hypotenuse of

13

and one leg of length

5

. Find the other leg.

Find the missing leg x.

Example 3

easy

Find the distance between

(1,2)

and

(4,6)

Example 4

medium

A right triangle has legs

2

and

5

. Find the exact length of the hypotenuse.

Find exact hypotenuse c.

Example 5

medium

A square has diagonal

10

. Find its side length.

Square diagonal = 10; find side s.

Example 6

hard

A right triangle has legs

a

and

a+7

and hypotenuse

a+8

. Find

a

Example 7

hard

In right triangle

ABC

with the right angle at

C

, leg

AC=5

and hypotenuse

AB=13

. Find leg

BC

and the area.

Find leg BC; 5-12-13 triple.

Example 8

challenge

Given a

3

–

4

–

5

right triangle, the altitude from the right angle hits the hypotenuse at point

D

. Find the two segments of the hypotenuse.

Practice Problems

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

Example 1

hard

A ladder

10

m long leans against a wall. The foot of the ladder is

6

m from the base of the wall. How high up the wall does the ladder reach?

Ladder = hypotenuse; find height h.

Example 2

hard

A right triangle has legs of length

x + 1

and

x + 8

, and hypotenuse

13

. Find

x

Example 3

easy

A right triangle has legs

6

and

8

. Find the hypotenuse.

Find hypotenuse c.

Example 4

easy

A right triangle has hypotenuse

13

and one leg

5

. Find the other leg.

Find leg b.

Example 5

easy

A right triangle has legs

3

and

4

. Find the hypotenuse.

Classic 3-4-5 triple: find c.

Example 6

easy

A right triangle has legs

9

and

12

. Find the hypotenuse.

Find hypotenuse c.

Example 7

easy

A square has side

6

. Find the length of its diagonal.

Square diagonal = hypotenuse; find d.

Example 8

easy

Is a triangle with sides

7

24

25

a right triangle?

Check: 7² + 24² = 25²?

Example 9

easy

Find the distance between points

(0, 0)

and

(3, 4)

in the plane.

Example 10

easy

A ladder

10

ft long leans against a wall with its base

6

ft from the wall. How high does it reach?

Find height h the ladder reaches.

Example 11

medium

Find the distance between

(-2, 1)

and

(4, 9)

Example 12

medium

Find the height of an equilateral triangle with side length

10

Find altitude h; answer is 5√3.

Example 13

medium

A rectangle has length

8

and width

6

. Find the length of its diagonal.

Rectangle diagonal d = hypotenuse.

Example 14

medium

A right triangle has hypotenuse

17

and one leg

8

. Find the other leg and the perimeter.

Find leg b; 8-15-17 triple.

Example 15

medium

Find the length of a diagonal of a rectangular box with edges

3

4

, and

12

Example 16

medium

A baseball diamond is a square with sides

90

ft. How far does the catcher (at home) throw to second base (diagonally opposite)?

Example 17

medium

Triangle has sides

5

7

9

. Is it a right triangle? If not, is the largest angle acute or obtuse?

Example 18

medium

A right triangle has legs

a

and

a + 1

and hypotenuse

a + 2

. Find

a

Example 19

medium

Triangle ABC has

A = (0, 0)

B = (5, 0)

C = (5, 12)

. Find

AC

Example 20

challenge

A right triangle has integer sides and one leg equal to

20

. List all possible hypotenuse lengths.

Example 21

challenge

In right triangle

ABC

with right angle at

C

AC = 6

and

BC = 8

. Find the length of the altitude from

C

to hypotenuse

AB

Example 22

challenge

Two ladders,

20

ft and

30

ft long, lean against opposite walls of an alley, crossing each other. The point where they cross is

8

ft above the ground. How wide is the alley? (Hard version — feel free to set up and observe.)

Example 23

easy

A right triangle has legs

5

and

12

. Find the hypotenuse.

Find hypotenuse c; 5-12-13 triple.

Example 24

easy

A right triangle has legs

8

and

15

. Find the hypotenuse.

Find hypotenuse c; 8-15-17 triple.

Example 25

easy

A right triangle has legs

1

and

1

. Find the hypotenuse in simplest radical form.

Find c in simplest radical form.

Example 26

easy

A TV screen is described as

16

inches wide and

12

inches tall. Find the diagonal.

TV screen: find diagonal d.

Example 27

medium

A right triangle has legs

9

and

40

. Find the hypotenuse.

Find hypotenuse c; 9-40-41 triple.

Example 28

medium

A right triangle has legs

6

and

8

. Find the length of the altitude drawn to the hypotenuse.

Example 29

medium

A rectangular field is

30

m by

40

m. Find the length of the diagonal path across it.

Find diagonal d across the field.

Example 30

medium

13

-ft ladder leans against a wall, reaching

12

ft up. How far is the ladder's foot from the wall?

Find base distance x.

Example 31

medium

In a right triangle, one leg is twice the other and the hypotenuse is

\sqrt{45}

. Find the legs.

Example 32

medium

Find the perimeter of a right triangle with legs

9

and

12

Find perimeter = 9 + 12 + 15.

Example 33

hard

A telephone pole is

24

ft tall. A support wire is anchored

7

ft from the base. How long is the wire?

Find wire length w; 7-24-25 triple.

Example 34

hard

A rectangular room is

12

ft long,

9

ft wide, and

8

ft tall. Find the space-diagonal length corner to opposite corner.

Example 35

hard

A boat sails

9

km east then

12

km north. How far is it from its starting point?

Find displacement d; 9-12-15 triple.

Example 36

hard

A right triangle has perimeter

30

and hypotenuse

13

. Find its legs.

Example 37

challenge

In a right triangle, the legs differ by

7

and the hypotenuse is

17

. Find the legs.

← Back to Pythagorean Theorem Formula Explained Practice Mode

Related Concepts

Triangles Exponents Square Roots Distance Formula Trigonometric Functions

Background Knowledge

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

trianglesexponentssquare roots