Pythagorean theorem

The Pythagorean Theorem was named after famous Greek mathematician Pythagoras. It is an important formula that states the following: a2 + b2 = c2

Pythagorean theorem

Looking at the figure above, did you make the following important observation? This may help us to see why the formula works.

  • The red square has 2 triangles in it 
  • The blue square has also 2 triangles in it
  • The black square has 4 of the same triangle in it
Pythagorean theorem

Therefore, area of red square  + area of blue square = area of black square

Let a = the length of a side of the red square

Let b = the length of a side of the blue square

Let c = the length of a side of the black square

Therefore, a2 + b2 = c2

Generally speaking, in any right triangle, let c be the length of the longest side (called hypotenuse) and let a and b be the length of the other two sides (called legs).

Right triangle
The theorem states that the length of the hypotenuse squared is equal to the length of side a squared plus the length of side b squared. Written as an equation, c2 = a2 + b2

Thus, given two sides, the third side can be found using the formula.

We will illustrate with examples, but before proceeding, you should know how to find the square root of a number and how to solve equations using subtraction.

Examples showing how to use the Pythagorean theorem


Exercise #1

Let a = 3 and b = 4. Find c, or the longest side

c2 = a2 + b2

c2 = 32 + 42

c2= 9 + 16

c2 = 25

c = √25

The sign (√) means square root

c = 5

Exercise #2

Let c = 10 and a = 8. Find b, or the other leg.

c2 = a2 + b2

102 = 82 + b2

100 = 64 + b2

100 - 64 = 64 - 64 + b2 (minus 64 from both sides to isolate b2 )

36 = 0 + b2

36 = b2

b = √36 = 6

Exercise #3

Let c = 13 and b = 5. Find a

c2 = a 2+ b2

132 = a2 + 52

169 = a2 + 25

169 - 25 = a2 + 25-25

144 = a2 + 0

144 = a2

a = √144 = 12

Take the Pythagorean theorem quiz below to see how well you understand this lesson.


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