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Rule

Area of a Rectangle

The area of a rectangle is the product of the rectangle's length and width

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.

Rectangle of length l and width w

Proof

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 and some other integer width into unit squares.
Rectangle divided in unit squares
Since the original rectangle has a side length and a width there are exactly rows of unit squares, each containing squares. This means that the total number of unit squares that make up the rectangle is the product of and
The area of the rectangle can be found by multiplying the number of unit squares by the area of one unit square,
The formula for the area of a rectangle of side length and width has been proven.

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