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What Is Pythagoras' Theorem?

What Is Pythagoras' Theorem?

Pythagorasí theorem is a theorem in geometry concerning the three sides of a right triangle.

Pythagoras' theorem (or the Pythagorean theorem) states that:
The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
It is easier to remember Pythagoras' theorem as a formula:



In the formula, c is the length of the hypotenuse (the longest side, opposite the right angle) and a and b are the lengths of the other two, shorter sides. The image below shows what we mean:

Visualizing Pythagoras' Theorem

The square of a side of a right triangle can be visualised by drawing a square on that side.

Pythagoras' theorem says that the area of the square on the longest side is equal to the sum of the areas of the squares on the other two sides.



By Pythagoras' theorem, the area of the red square is equal to the areas of the blue and green squares added together.

The slider below helps visualize Pythagoras' theorem:

A Real Example of Pythagoras' Theorem


An Example Question

The image below shows a right triangle with sides of 3, 4 and 5.



Show that Pythagoras' theorem works for this right triangle.
Step 1
Start with the formula for Pythagoras' theorem:
a2 + b2 = c2
Don't forget: a2 = a × a (a squared) and b2 = b × b and c2 = c × c.

Step 2
Find the lengths of the sides from the right triangle.

In our example, the two shorter side lengths are a = 3 and b = 4. The longest side length is c = 5.

Step 3
Substitute a = 3, b = 4 and c = 5 into the formula.
32 + 42 = 52
(3 × 3) + (4 × 4) = 5 × 5
9 + 16 = 25
9 + 16 does equal 25. Pythagoras' theorem works.

Did you know?: a 3,4,5 triangle is known as a Pythagorean triple. All three sides are integers (see Note).

How to Use Pythagoras' Theorem

Pythagoras' theorem relates the three sides of a right triangle to each other. If two of the sides are known, the third unknown side can be found:
  • Find the hypotenuse. If the shorter two sides are known, Pythagoras' theorem can find the hypotenuse:

    An Example Question

    What is the hypotenuse of the right triangle below?

    a = 3 and b = 4. Find c.

    Use Pythagoras' theorem, a2 + b2 = c2, and rearrange:
    a2 + b2 = c2
    c2 = a2 + b2
    c2 = 32 + 42 = (3 × 3) + (4 × 4) = 9 + 16 = 25
    c = √c2 = √25
    c = 5
    The hypotenuse is 5.

    Read more about how to find the hypotenuse using Pythagoras' theorem.

  • Find a shorter side. If the hypotenuse and one of the shorter sides are known, Pythagoras' theorem can find the other shorter side:

    An Example Question

    What is the missing side of the right triangle below?

    b = 4 and c = 5. Find a.

    Use Pythagoras' theorem, a2 + b2 = c2, and rearrange:
    a2 + b2 = c2
    a2 = c2 - b2
    a2 = 52 - 42 = (5 × 5) - (4 × 4) = 25 - 16 = 9
    a = √a2 = √9
    a = 3
    The missing side is 3.

    Read more about how to find a shorter side using Pythagoras' theorem.
Interactive Widget
Here is an interactive widget to help you learn about Pythagoras' theorem.
Interactive Test
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Here's a second test on Pythagoras' theorem.
Here's a third test on Pythagoras' theorem.
Note

Who Was Pythagoras?



Pythagoras was an ancient Greek philosopher, mathematician and founder of a religious movement, who lived from about 570 - 495 BC.

His most famous contribution is the theorem discussed here.

He started a religious movement, whose followers called themselves Pythagoreans. They believed numbers were behind all things, giving them mystic importance.

They also believed in the transmigration of souls - that a person's soul would move to another person, animal or vegetable after they died.

Pythagorean Triples

A Pythagorean triple is a right triangle where the length of each side is an integer (a whole number).

The most common Pythagorean triple is the 3,4,5 triangle:


32 + 42 = 52
9 + 16 = 25
All multiples of 3, 4 and 5 these are also Pythagorean triples:
6,8,10
9,12,15
12,16,20
Other Pythagorean triples are 5,12,13 and 7,24,25.

Pythagoras' Theorem Only Works For Rigth Triangles

In the film The Wizard of Oz, the straw-headed Scarecrow goes to the Wizard to get a brain.

At the end of the film, the Wizard awards him a Diploma as proof that he is now brainy.



On receiving the Diploma, the Scarecrow says:
"The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh joy, rapture! Iíve got a brain!"
The Wizard was obviously kidding the Scarecrow.

If he really did have a brain, he would realize Pythagoras' theorem applied to right triangles, not isosceles triangles, and that it is the square of the sides, not the square root.