The Pythagorean Theorem is a very handy way to find the length of any one side of a right triangle if you know the length of the other two sides.

Table of Contents

  1. What is the Pythagorean Theorem?
  2. Pythagorean Theorem Formula
  3. How To Use The Pythagorean Theorem
  4. Pythagorean Theorem With Square Roots
  5. Pythagorean Theorem Word Problems
  6. Pythagorean Theorem Examples
  7. Pythagorean Triples
  8. Pythagorean Theorem Proof

What is the Pythagorean Theorem?

The Pythagorean Theorem states that in right triangles, the sum of the squares of the two legs (a and b) is equal to the square of the hypotenuse (c).

Pythagorean Theorem Definition

Pythagorean Theorem History

The Pythagorean Theorem is named after and written by the Greek mathematician, Pythagoras. Pythagoras is pronounced ("pi-thag-uh-rus," with a short "I" sound in his first syllable; pi as in pin), but the theorem has been described in many civilizations worldwide. The theorem is pronounced "pi-thag-uh-ree-uhn."

A theorem in geometry is a provable statement. The Pythagorean Theorem was proven very long ago.

Pythagorean Theorem Formula

In any right triangle ABC, the longest side is the hypotenuse, usually labeled c and opposite ∠C. The two legs, a and b, are opposite A and B. C is a right angle, 90°, and A + B = 90° (complementary).

The three sides always maintain a relationship such that the sum of the squares of the legs is equal to the square of the hypotenuse. In mathematics terms it looks like this:

a2 + b2 = c2

Pythagorean Theorem Formula

How To Use The Pythagorean Theorem

The Pythagorean Theorem can be used to find the length of the hypotenuse if you know the lengths of legs a and b.

Solve for c

Suppose you have leg a = 5 centimeters and b = 12 centimeters:

Pythagorean Theorem - Solve For c

Pythagorean Theorem Steps Explained

Let's start with our formula:

a2 + b2 = c2

Then we plug in the length of each leg:

52 + 122 = c2

Multiply each number times itself:

25 + 144 = c2

Then we add:

169 = c2

Now, we take the square root of both sides:

169 = c2

So you take the principal root of both sides and you get:

13 = c

Solve for a or b

You can know the length of the hypotenuse and one leg and still use the Pythagorean Theorem. Suppose you know c = 40 feet and short leg a = 24 feet.

Pythagorean Theorem - Solve for b

Step By Step

Our formula:

a2 + b2 = c2

First, plug in what you know:

242 + b2 = 402

Multiply each number times itself, then add:

576 + b2 = 1600

Then, you need to subtract the a2 length from both sides, to isolate b2:

(576 - 576) + b2 = (1600 - 576)

On the left handed side we're left with just b2.

b2 = 1024

Now take the square root of both sides:

b2 = 1024

Take the principal root of both sides and you get:

b = 32

Checking your answer

So our right triangle had legs a = 24, b = 32, and hypotenuse c = 40. If you do not believe your answer, plug all three numbers back into the Pythagorean Theorem:

242 + 322 = 402

Let's check if the numbers add up:

576 + 1024 = 1600

Everything checks out; we were right! And our numbers were nice, whole integers, which made the work easy.

Pythagorean Theorem With Square Roots

You can use the Pythagorean Theorem to solve for any length if you know the lengths of two other sides.

Suppose you need the length of the hypotenuse c. Then you simply need the square root of the sum of a2 + b2, like this:

c = a2 + b2

If you need to find the short leg a, manipulate the formula to look like this:

a = c2 - b2

And if you need to find the longer leg b, you rewrite the formula to look like this:

b = c2 - a2

Pythagorean Theorem Word Problems

An firefighter's extension ladder is leaning against a building, so its top just touches the gutters at the roof edge. You know the ladder is 41 feet long and it is 9 feet from the wall of the building. How high is the building?

Pythagorean Theorem Word Problems

The ladder is the hypotenuse, 41', and leg a is the short leg, 9'. Plug in what you know into either formula you wish to use to solve for long leg b:

Start with our formula:

a2 + b2 = c2

First, plug in what you know:

92 + b2 = 412

Multiply each number times itself, then add:

81 + b2 = 1681

Then, you need to subtract the a2 length from both sides, to isolate b2:

(81 - 81) + b2 = (1681 - 81)

On the left handed side we're left with just b2.

b2 = 1600

Now take the square root of both sides:

b2 = 1600

Take the principal root of both sides and you get:

b = 40

Let's solve this another way!

If you need to find the longer leg b, you rewrite the formula to look like this:

b = c2 - a2

b = 412 - 92

b = 1681 - 81

b = 1600

b = 40

Pythagorean Theorem Examples

Find the answers to these five Pythagorean Theorem problems:

Example #1

Find the hypotenuse c for a right triangle with the length of short leg a = 6 and the length of the long leg b = 8:

a2 + b2 = c2

62 + 82 = c2 

36 + 64 = c2 

100 = c2 

100 = c2 

10 = c

Example #2

Find the short leg a for a right triangle with the length of long leg b = 24 and the length of the hypotenuse c = 25:

a2 + b2 = c2

a2 + 242 = 252 

a2 + 576 = 625

a2 + (576 - 576) = (625 - 576)

a2 = 49

a2 = 49

a = 7

Example #3

Find the long leg b for a right triangle with the length of short leg a = 65 and the length of the hypotenuse c = 97:

a2 + b2 = c2

652 + b2 = 972

4,225 + b2 = 9,409

(4,225 - 4,225) + b2 = (9,409 - 4,225)

b2 = 5,184

b2 = 5,184

b = 72

Example #4

Find the short leg a for a right triangle with the length of long leg b = 60 and the length of the hypotenuse c = 68:

a = c2 - b2

a = 682 - 602

a = (68 + 60) (68 - 60)

b = (128 × 8)

a = 10242

a = 32

Example #5

Find the long leg b for a right triangle with the length of short leg a = 60 and the length of the hypotenuse c = 100:

b = c2 - a2

b = 1002 - 602

b = (100 + 60) (100 - 60)

b = (160 × 40)

b = 6,4002

b = 80

Pythagorean Triples

The reason our example problems ended up with nice, neat, whole numbers is because we used Pythagorean Triples, or three whole numbers that work to fulfill the Pythagorean Theorem.

The smallest Pythagorean Triple is 3, 4, 5 (a right triangle with legs of 3 and 4 units, and a hypotenuse of 5 units). All the multiples of that triple will also be triples:

6, 8, 10

9, 12, 15

12, 16, 20

The list never ends, and includes one of our examples: 24, 32, 40. Other Pythagorean Triples exist, too:

5, 12, 13 (we used that one in our example!)

8, 15, 17

9, 40, 41

That list never ends, either!

Learn how to identify and categorize Pythagorean Triples as either primitive or imprimitive (not primitive) and learn how to use the Pythagorean Theorem to find Pythagorean triples.

Pythagorean Theorem Proof

Thousands of proofs of this theorem exist, including one by U.S. president James Garfield (before he became president). One proof is easy to make with graph paper, a straightedge, pencil, and scissors.

Construct ABC with legs a and b to the left and bottom and hypotenuse c at the top right. Leg a is opposite A, leg b is opposite B, and hypotenuse c is opposite right angle C.

Let length a = 3, b = 4, and hypotenuse c = 5.

Right Angled Triangle

How To Prove The Pythagorean Theorem

Construct a square using leg a as the right side of the square. It will be 9 square units (a2). Construct a square using leg b as the top side of its square, so it is 16 square units (b2). Cut out another 5 x 5 square and line it up with hypotenuse c, so the square is c2.

Think: what is 9 square units + 16 square units? It is 25 square units, the area of c2.

Pythagorean Theorem Proof

area = a × a = a2

area = b × b = b2

area = c × c = c2

a2 + b2 = c2

Lesson Summary

If you worked carefully, now you should know what the Pythagorean Theorem is, recognize it when you see it, and apply it to solve problems in geometry. You can use it to find the length of any one side of a right triangle if you know the lengths of the other two sides. We also learned how to prove the Pythagorean Theorem.

Next Lesson:

Converse of The Pythagorean Theorem

What you learned:

Once you work your way through the great video, examine the graphics, and read the information here, you will learn to:

  • Recognize and apply the Pythagorean Theorem
  • Use the Pythagorean Theorem to calculate the length of one side of a right triangle if you know the lengths of the other two sides.
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
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Ashburn, VA

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