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Rule

Area of a Rectangle

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

ℓ

and width

w .

$A = ℓ w$

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.

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

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 \times 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 \times w) \times 1 = ℓ w

The formula for the area of a rectangle of side length

ℓ

and width

w

has been proven.

$A = ℓ w$

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

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Proof