SectionA.1Rectangle and Square
The rectangle on the left has a pair of lengths (top and bottom) and a pair of widths (left and right). The square on the right simply has four side lengths of equal size. A square can be considered a special rectangle whose length and width have the same size.
Perimeter is the distance around a two-dimensional shape. To calculate perimeter, start from a point on the shape (usually a corner), travel around the shape, and calculate the sum of the distance traveled.
For the rectangle in the figure, if we travel around it, the total distance would be:
\begin{align*}
\text{rectangle perimeter}\amp=3\text{ cm}+2\text{ cm}+3\text{ cm}+2\text{ cm}\\
\amp=10\text{ cm}\text{.}
\end{align*}
Since a rectangle has two pairs of equal sides, we could have calculated the perimeter this way:
\begin{align*}
\text{rectangle perimeter}\amp=2(3\text{ cm}+2\text{ cm})\\
\amp=2(5\text{ cm})\\
\amp=10\text{ cm}\text{.}
\end{align*}
It's even easier to calculate the square's perimeter:
\begin{align*}
\text{square perimeter}\amp=4(2\text{ cm})\\
\amp=8\text{ cm}\text{.}
\end{align*}
Let's summarize the perimeter formulas of rectangle and square:
- Rectangle
\(P=2(\ell+w)\text{,}\) where \(P\) is the rectangle's perimeter, \(\ell\) is its length, and \(w\) is its width.
- Square
\(P=4\ell\text{,}\) where \(P\) is the square's perimeter and \(\ell\) is its length.
Area of a two-dimensional shape is the number of unit squares that can be contained within it (possible after morphing them into non-square shapes). A unit square is a square with a side length of \(1\) unit. In Figure A.1.1, the rectangle has \(6\) unit squares in it, so its area is \(6\) square centimeters; the square has \(4\) unit square in it, so its area is \(4\) square centimeters.
To calculate the area of a rectangle, we usually use multiplication. The rectangle in Figure A.1.1 has two rows of \(3\) unit squares, so its area is \(3\text{ cm}\cdot2\text{ cm}=6\text{ cm}^2\text{.}\) In general, the area formula of rectangles and squares are:
- Rectangle
\(A=\ell w\text{,}\) where \(A\) is the rectangle's area, \(\ell\) is its length, and \(w\) is its width.
- Square
\(A=\ell^2\text{,}\) where \(A\) is the square's area and \(\ell\) is its length.
Note that just as \(3\cdot3=3^2\text{,}\) we can treat units in the same way: \(\text{cm}\cdot\text{cm}=\text{cm}^2\text{.}\) The area unit reads "square centimeters," which is different from the length unit \(\text{cm}\text{.}\) In application problems, you are encouraged to include units in calculations.