Area of a Rectangle

Consider the unit square, a square with side lengths of one unit. By the definition of area, the space inside the unit square is one square unit. Now, divide a rectangle of some integer length $\ell$ and some other integer width $w$ into unit squares. Since the original rectangle has a side length $\ell$ and a width $w,$ there are exactly $w$ rows of unit squares, each containing $\ell$ squares. This means that the total number of unit squares that make up the rectangle is the product of $\ell$ and $w.$ The area of the rectangle $A$ can be found by multiplying the number of unit squares by the area of one unit square, $1.$ The formula for the area of a rectangle of side length $\ell$ and width $w$ has been proven.

Note that this result is still valid if $\ell$ and $w$ were any real numbers.