Area of a Rectangle
The area of a rectangle is found by multiplying its length by its width, giving the number of square units it covers.
Formula
A = l \times w
Definition
To find the area of a rectangle, multiply its length by its width, giving an answer in square units: $A = l \cdot w$, and for a square with side $s$, $A = s^2$. This formula counts the number of unit squares that tile the interior with no gaps or overlaps, and it is the foundation for every other area formula, since the parallelogram, triangle, and trapezoid formulas are all derived by rearranging or halving a rectangle. Formally, the area of a rectangle is the Lebesgue measure of the Cartesian product $[0, l] \times [0, w]$ in $\mathbb{R}^2$, equal to $l \cdot w$ by Fubini's theorem (the double integral factors as the product of two single integrals), and scaling one dimension by factor $k$ scales the area by $k$.
Example
A rectangle $7$ cm long and $4$ cm wide has $A = 7 \times 4 = 28$ cm², like laying $7$ rows of $4$ tiny squares inside it. A room $12$ ft by $9$ ft has area $108$ ft$^2$, so at $\$3$ per ft$^2$ of carpet the total cost is $\$324$; doubling the length doubles the area, while doubling both dimensions quadruples it. Under a general linear transformation with matrix $M$, all areas scale by $|\det(M)|$, so a rectangle scaled by $\text{diag}(a, b)$ has area $a \cdot b$ times the original, a foundational result in linear algebra and multivariable calculus.
Key Insight
A square is just a special rectangle where length equals width, so $A = s \times s = s^2$. All polygon area formulas reduce to the rectangle formula under decomposition or shear. The determinant of a $2 \times 2$ matrix equals the signed area of the parallelogram spanned by its column vectors, connecting the rectangle area formula directly to the algebraic concept of determinants.