Rectangles: Perimeter, Area, and Missing Sides

Start with the distance around. The perimeter of a figure is the total distance around it. Picture walking once around the edge and measuring how far you went. To find a perimeter, you add the lengths of all the sides. The lengths must be in the same unit, and the perimeter is in that same unit of length.

The figure below shows a rectangle on a coordinate plane. The horizontal axis is the $x$-axis and the vertical axis is the $y$-axis. Each point is named by a pair $(x, y)$: $x$ tells how far across the point is and $y$ tells how far up. One grid step stands for $1$ centimeter. The corners of the rectangle, called vertices, are at $(1, 1)$, $(7, 1)$, $(7, 5)$ and $(1, 5)$.

The two bottom vertices share the $y$-value $1$, and the two top vertices share the $y$-value $5$. So the bottom and top sides are horizontal. The two left vertices share the $x$-value $1$, and the two right vertices share the $x$-value $7$. So the left and right sides are vertical. Horizontal and vertical grid lines are perpendicular, which means they meet at right angles. This figure has four right angles because of where its vertices are, not because of how it looks.

That shortcut works for every rectangle, because opposite sides of a rectangle are always equal. Let $l$ stand for the length of one side and $w$ for the length of a side next to it, both in the same unit. People often call $l$ the length and $w$ the width. Going once around, you pass $l$, $w$, $l$ and $w$, so the perimeter $P$ is $l + w + l + w$. You can group those four sides as $(l + w) + (l + w)$. That is two copies of $l + w$, and regrouping does not change the total. So $P = 2 \times (l + w)$.

A square is a rectangle with $l$ and $w$ equal. Call its side length $s$. Its perimeter is $s + s + s + s$, which is $4 \times s$. For example, a square with sides of $5$ centimeters has a perimeter of $4 \times 5 = 20$ centimeters. The rectangle rule agrees: $2 \times (5 + 5) = 20$.

The area of a figure is the amount of flat surface it covers. You measure area by counting unit squares. A unit square has sides $1$ unit long, and the unit squares must cover the figure with no gaps and no overlaps. A square with sides of $1$ centimeter has an area of $1$ square centimeter, also written $1 \text{ cm}^2$.

The figure below shows the same rectangle cut into four rows. The bottom row has vertices $(1, 1)$, $(7, 1)$, $(7, 2)$ and $(1, 2)$. Its sides are horizontal and vertical, so it is a rectangle too. It is $7 - 1 = 6$ centimeters long and $2 - 1 = 1$ centimeter tall. On this grid each cell is one unit square, so the row holds $6$ unit squares. The other three rows lie between the $y$-values $2$ and $3$, $3$ and $4$, and $4$ and $5$. Each of them also runs from the $x$-value $1$ to the $x$-value $7$, so each is $6$ centimeters long and $1$ centimeter tall.

The four rows cover the whole rectangle with no gaps and no overlaps. So the rectangle holds $4 \times 6 = 24$ unit squares, and its area is $24$ square centimeters.

Notice where the $4$ and the $6$ come from. The rectangle is $6$ centimeters long, so each row holds $6$ unit squares. It is $4$ centimeters wide, so there are $4$ rows. The $4$-centimeter side is perpendicular to the $6$-centimeter side, so the rows stack straight up and fill the rectangle exactly. That is why you can find the area $A$ by multiplying the two side lengths instead of counting: $A = 6 \times 4 = 24$ square centimeters.

A square works the same way, with both side lengths equal. A square with sides of $5$ centimeters holds $5$ rows of $5$ unit squares, so its area is $5 \times 5 = 25$ square centimeters. Earlier you found that its perimeter is $20$ centimeters. The two measures answer different questions, so they use different units.

Sometimes you know the area or the perimeter of a rectangle, but one side length is missing. You can still find it. Write the rule as an equation, which is a statement that two amounts are equal. Use a letter for the side length you do not know. Then undo the operations one at a time until the letter stands alone.

To undo an operation, use its inverse operation. Dividing by a nonzero number undoes multiplying by it, and subtracting a number undoes adding it. An equation stays true when you divide both sides of the equation by the same nonzero number, or subtract the same number from both sides. The two amounts change in the same way, so they stay equal.

A square is quicker to work backward on, because its four sides are equal. Suppose a square has a perimeter of $36$ centimeters. Let $s$ be its side length in centimeters. Then $4 \times s = 36$. Divide both sides of the equation by $4$ to get $s = 36 \div 4 = 9$. Each side is $9$ centimeters long. Check: $4 \times 9 = 36$.

Here is the idea to keep. The perimeter of a rectangle is the distance around it, so you add: $P = 2 \times (l + w)$, in units of length. The area is the surface it covers, so you multiply two sides that meet at a right angle: $A = l \times w$, in square units. Both rules need side lengths in the same unit, and a square follows both with equal sides. When a side length is missing, write the rule as an equation and undo each operation in turn.

Next, you will use these ideas to solve area and perimeter word problems and to compare the area and perimeter of two figures.