Algebra I

Pythagorean Theorem

Lesson

In a right triangle (one with a 90° angle), the two shorter sides are called the legs, and the longest side — directly across from the right angle — is called the hypotenuse.

The Pythagorean theorem says that for any right triangle, if the legs are $a$ and $b$ and the hypotenuse is $c$ :

a^2 + b^2 = c^2

This means: square each leg, add them up, and you get the square of the hypotenuse. To find a missing side, plug in what you know and solve.

Worked example 1 — finding the hypotenuse

Legs are 3 and 4. Find the hypotenuse $c$ .

3^2 + 4^2 = c^2

9 + 16 = c^2

25 = c^2

Take the positive square root:

c = 5

Worked example 2 — finding a leg

One leg is 6 and the hypotenuse is 10. Find the other leg $b$ .

6^2 + b^2 = 10^2

36 + b^2 = 100

Subtract 36 from both sides:

b^2 = 64

b = 8

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

A right triangle has legs 3 and 4. Find the hypotenuse.

Problem 2

A right triangle has legs 6 and 8. Find the hypotenuse.

Problem 3

A right triangle has legs 5 and 12. Find the hypotenuse.

Practice

Standard problems matching the lesson.

Problem 4

A right triangle has legs 8 and 15. Find the hypotenuse.

Problem 5

A right triangle has legs 7 and 24. Find the hypotenuse.

Problem 6

A right triangle has legs 9 and 12. Find the hypotenuse.

Problem 7

A right triangle has one leg 6 and hypotenuse 10. Find the other leg.

Problem 8

A right triangle has one leg 9 and hypotenuse 15. Find the other leg.

Problem 9

A right triangle has one leg 5 and hypotenuse 13. Find the other leg.

Problem 10

A right triangle has one leg 8 and hypotenuse 17. Find the other leg.

Problem 11

A right triangle has legs 20 and 21. Find the hypotenuse.

Problem 12

A right triangle has one leg 11 and hypotenuse 61. Find the other leg.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 13

A right triangle has legs 9 and 40. Find the hypotenuse.

Problem 14

A right triangle has legs 12 and 35. Find the hypotenuse.

Problem 15

A right triangle has one leg 28 and hypotenuse 53. Find the other leg.

Problem 16

A right triangle has one leg 33 and hypotenuse 65. Find the other leg.

Problem 17

A right triangle has legs 14 and 48. Find the hypotenuse.

Problem 18

A right triangle has legs 15 and 36. Find the hypotenuse.

Practice

Standard problems matching the lesson.

Problem 19

A 13-ft ladder leans against a wall, base 5 ft from the wall. How high up the wall does the top reach (ft)?

Problem 20

A TV's diagonal is 50 in and its width is 40 in. What is its height (in)?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 21

A guy wire runs from the top of a 24-ft pole to a stake 7 ft from the base. How long is the wire (ft)?

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Quiz

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