Area of a Rectangle
Geometry & MeasurementThe 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 times its width. The answer is in square units.
Example
A rectangle $7$ cm long and $4$ cm wide: $A = 7 \times 4 = 28$ cm². Imagine laying $7$ rows of $4$ tiny squares inside the rectangle; you get $28$ squares total.
Key Insight
The rectangle area formula is the foundation for all other area formulas. A square is just a special rectangle where length equals width, so $A = s \times s = s^2$.
Definition
The area of a rectangle with length $l$ and width $w$ is $A = l \cdot w$. 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.
Example
A room $12$ ft by $9$ ft has area $108$ ft$^2$. If carpet costs $\$3$ per ft$^2$, the total cost is $\$324$. Doubling the length doubles the area; doubling both dimensions quadruples the area.
Key Insight
All polygon area formulas reduce to the rectangle formula under decomposition or shear. The parallelogram, triangle, and trapezoid formulas are all derived by rearranging or halving a rectangle.
Definition
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). Scaling one dimension by factor $k$ scales the area by $k$.
Example
A rectangle scaled by matrix $\text{diag}(a, b)$ has area $a \cdot b$ times the original. Under a general linear transformation with matrix $M$, all areas scale by $|\det(M)|$, a foundational result in linear algebra and multivariable calculus.
Key Insight
The determinant of a $2 \times 2$ matrix equals the signed area of the parallelogram spanned by its column vectors. This connects the rectangle area formula directly to the algebraic concept of determinants.