Years ago, a man named Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90°), and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together.

It is called “Pythagoras’ Theorem” and can be written in one short equation:

a2 + b2 = c2



  • c is the longest side of the triangle
  • a and b are the other two sides

The longest side of the triangle is called the “hypotenuse”, so the formal definition is:

In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let’s see if it really works using an example.


A “3,4,5” triangle has a right angle in it.

pythagoras theorem Let’s check if the areas are the same:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

It works … like Magic!

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)


How Do I Use it?

Write it down as an equation:

abc triangle a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples:

Example: Solve this triangle.

right angled triangle  

a2 + b2 = c2

52 + 122 = c2

25 + 144 = c2

169 = c2

c2 = 169

c = √169

c = 13


You can also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

right angled triangle  

a2 + b2 = c2

92 + b2 = 152

81 + b2 = 225

Take 81 from both sides:

b2 = 144

b = √144

b = 12

Example: What is the diagonal distance across a square of size 1?

Unit Square Diagonal  

a2 + b2 = c2

12 + 12 = c2

1 + 1 = c2

2 = c2

c2 = 2

c = √2 = 1.4142…


It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.

Example: Does this triangle have a Right Angle?

10 24 26 triangle Does a2 + b2 = c2 ?

  • a2 + b2 = 102 + 242 = 100 + 576 = 676
  • c2 = 262 = 676

They are equal, so …

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

Does 82 + 152 = 16?

  • 82 + 152 = 64 + 225 = 289,
  • but 16256

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Triangle with roots Does a2 + b2 = c2 ?

Does (3)2 + (5)2 = (8)2 ?
Does 3 + 5 = 8 ?

Yes, it does!

So this is a right-angled triangle



11 comments on “PYTHAGORAS’ THEOREM

  1. Sweet Honey says:

    Tnx for tis example question….

  2. i like the example question .. but i’m low in mathemtics .. i think i can’t answer the simple question ..
    i don’t know why i hate mathematics too much ..

  3. Annber lynn says:

    i try to understand this formula..any other easy ways?

    • as far as i know, this is the only formula for pythagoras’ theorem..let me explain in a simpler way..hopefully i can help

      first thing u need to know is this formula can be applied for right angled triangle only (has 90degree angle)
      then u must know which one is the hypotenuse (the opposite side of the 90degree angle)..hypotenuse is the longest side
      then apply the formula

      a2 + b2 = c2
      c is the hypotenuse, a and b is the other two sides

      if the value of hypotenuse is given, and we want to find the value of either a or b, change the formula
      c2- a2 = b2


      c2- b2 = a2

      but u must remember to get the final answer, find the square root.. i hope u know what is square root

      i hope u understand my explanation..try the examples above 🙂

  4. Annber lynn says:


  5. Tulius says:

    can you teach me online…my exams is only 4 weeks away?

  6. aiden says:

    can you teach me because my exam is 12 days away? please………………….

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