Is there any logical proof or derivativation for area of a square

Question

Area of a square =a²

 

How was this formula derived ? 

Answer 1

the area of a rectangle (or any polygon) is the surface enclosed by the rectangle

area is measured in square units

the area of a rectangle (or any polygon), is the number of square units (square inches, square centimeters, square meters, square feet, etc) that can fit inside of the rectangle, side by side, non-overlapping, until the surface inside of the rectangle is completely covered

this is pretty simple if the length and width of the rectangle is measured in whole number units, such as inches

if you have a rectangle 5 inches long and 3 inches wide, you can completely cover the surface inside of the rectangle with 15 square inches (try this on a piece of graph paper)

the procedure becomes more difficult if the length and width are very large numbers or numbers that are not whole numbers, or, if the shape of the polygon is irregular

for these reasons, we have formulas to give us the answers instead of trying to physically do this

in the case of a square, if you found the area on a piece of graph paper you would see that the number of square units in each row is the same as the number of square units in each column

the area of the square is the number of square units in a row multiplied by the number of square units in a column

you can verify this by actually counting the total number of square units within the square itself

but it is much easier to use a formula

Area=# square units in a row times # of square units in a column

since in a square the number of rows is equal to the number of columns, we us a letter such as "s" to represent the measure of both the rows and columns, where s stands for "side" and the formula is A=s*s or A=s²

Answer 2

The area of a rectangle is l x w. Since a square is a rectangle, the formula is the same, but all 4 sides have the same length.