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Rule

$A=ℓw$

The length is typically defined as the measurement of the rectangle's longer sides, while the width refers to the measurement of its shorter sides; however, this assignment is arbitrary.

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 $ℓ$ and some other integer width $w$ into unit squares.

Since the original rectangle has a side length $ℓ$ and a width $w,$ there are exactly $w$ rows of unit squares, each containing $ℓ$ squares. This means that the total number of unit squares that make up the rectangle is the product of $ℓ$ and $w.$

$Number of Unit Squares:ℓ×w $

The area of the rectangle $A$ can be found by multiplying the number of unit squares by the area of one unit square, $1.$ $A=(ℓ×w)×1=ℓw $

The formula for the area of a rectangle of side length $ℓ$ and width $w$ has been proven. $A=ℓw$

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