Pythagorean Theorem: Definition & Example

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  • 0:03 The Pythagorean Theorem
  • 0:37 Right Triangles
  • 1:12 The Sides
  • 2:32 Application
  • 5:01 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Learn what the Pythagorean Theorem says about a right triangle and the relationships of its three sides to each other by watching this video lesson. Also learn how you can use this to your advantage when solving problems.

The Pythagorean Theorem

The Pythagorean Theorem is named after the Greek mathematician named Pythagoras who lived around 500 BC. Yes, he's old, but the theorem itself is even older. It is named after him because he is supposedly the first one to prove that the theorem is true. What does this theorem say? The Pythagorean Theorem tells us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides. In equation form, it is a^2 + b^2 = c^2.

Right Triangles

The three letters used in the equation form of the Pythagorean Theorem are specific to right triangles only. A right triangle is a triangle with exactly one right angle measuring 90 degrees. Take a square and cut it diagonally, and you will have two right triangles. The same goes for a rectangle. Cut a rectangle in half diagonally, and you are left with two right triangles. The only criterion for a right triangle is that it has one right angle. Just think right triangle equals right angle.

The Sides

Once you have your right triangle, you can label its sides appropriately with the letters. The letter c is always the hypotenuse, the side opposite the right angle and also the longest side. The other two letters, a and b, are the other two sides. It doesn't matter which side you name a and which side you name b as long as these two letters are naming these two sides. But once you name them, just make sure you keep your a and b sides straight, and don't get them mixed up.

Once labeled, you can see that the theorem is telling you that if you square all the sides, then the sum of sides a and b will equal that of side c, the hypotenuse. You can actually prove this to yourself. If you think about it, if you take the square of a side, you are actually finding the area of a square whose side is the same length as that side of the triangle. The area of a square is the square of one of its sides.

So if you drew three squares - one for each side of the triangle - you can cut them out. Then you can play around with the two smaller squares and make them fit into the larger one. Cut the smaller ones if needed, and you will see that the two smaller ones fit perfectly inside the largest square.


So, why is the Pythagorean Theorem such a big deal? It is a big deal because it is so useful. You can use it to find the length or distance of a missing side. In the construction field, right triangles can be found everywhere and if you know the measurements of two of the sides of the right triangle, then you can find the measurement of the third without measuring. This is very useful when drawing blueprints and such and you need a measurement. Want to see how the theorem works?

We start with a right triangle where we only know two of the sides. We label the triangle with our three letters. We see that our hypotenuse - side c- is the side that we are missing a measurement. Our side a measures 4 centimeters, and our side b measures 3 centimeters.

If we know sides A and B, we can find side C, or the hypotenuse
Right triangle with sides 4cm, 3cm, and ?

From this point, we can plug our information into the equation form of the Pythagorean Theorem.

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