Intuition behind Pythagoras Theorem

The Pythagoras Theorem states that in a right angled triangle, ‘a’ being the base, ‘b’ being the height and ‘c’ being the hypotenuse of  that triangle, then a²+b²=c²

Below is an illustration of this –

Example –

1. if the base of a right angled triangle is 3, the height is 4,then what is the length of its hypotenuse?
Solution –  given, a=3, b=4 ,c=?
          Using Pythagoras theorem, 
          a²+b²=c²
          3²+4²=c²
          √(9+16) =c
          c=5

2. if the hypotenuse of a right angled triangle is 13, the height is 5,then what is the length of its base?
Solution –
given, a=?, b=5 ,c=13    
Using Pythagoras theorem,          
a²+b²=c² 
a²+5²=13²
a=√(169-25)
a=12

Intuition behind Pythagoras Theorem :
Let’s prove this theorem using the figures. 
Draw squares corresponding to each side of the triangle as follows –

If we look at the figure closely, we could reframe the Pythagoras theorem as follows-
The area of 2 squares is equal to the third square.
ie- a² is the area of the 1st square
     b² is the area of the 2nd square
     c² is the area of the 3rd square

thus, a²+b²=c²
Another proof of Pythagoras theorem can be shown by rearranging the triangles to form 2 squares as follows

If we compare the two squares, we can find that both the squares have a+b side length, thus having the same area.
In each square, four right-angled triangles are used (realigned in a different way though)
So, we can conclude that

area(1st square) =a rea(2nd square)
c² + 4* (area of a right angled triangle)=  a²+b²+4* (area of a right angled triangle)
c²=a²+b²  [cancelling the common terms from both sides]

Thus, pythagoras theorem is proved.