Proof for area of a square

Question

Why area of a square =a2

A2
Can also mean a+a+a. . a times
Which could represent a linear addition of a side with "a" unit for "a" number of times .

If a line/ side is 3units , then 3x3 = 3+3+3 =9

How do we know that the value "9" for the above example is not actually the addition of 3 lines and in fact represents the area of square .

Answer 1

With a side length of 3, the area is 3² = 3(3) = 3+3+3 = 9

With a side length of 4, the area is 4² = 4(4) = 4+4+4+4 = 16

With a side length of 5 the area is 5² = 5(5) = 5+5+5+5+5 = 25

 

The area is NOT the addition of 3 lines... It is the side length multiplied

by itself.  S² = S(S) = S+S+S + ... + S    a total of S times

 

With a side lenth of 112 I would much rather do 112² than add 112 a total of 112 times to reach 12,544.