How to Find the Perimeter of a Shape

You already know the perimeter is the distance all the way around a figure, the total you get by adding up its sides. Now we make that quick. We will build formulas for rectangles and squares, deduce missing side lengths on figures that are only partly labelled, and see what cutting a piece out of a shape does to its boundary, which is sometimes nothing at all.

Problem

In the diagram above is a rectangle 12 units wide and 7 units tall. Find its perimeter. Give the number of units.

Show a hint

The perimeter is the distance all the way around. A rectangle has two widths and two heights, so the shortcut is $P = 2(l+w)$.
Add the width and the height first, then double that sum.

Show the full solution

Add the width and the height, then double. $$P = 2(l+w) = 2(12+7) = 2\times 19 = 38$$ The perimeter is $\boxed{38}$ units. Doubling works because a rectangle has two equal widths and two equal heights, so you never need to add all four sides.

Problem

In the diagram above is a square with side 9. Find its perimeter. Give the number of units.

Show a hint

A square has four sides, and they are all the same length. So instead of adding four different numbers, you can add the same number four times.
That repeated addition is just multiplication. Multiply the side length by 4.

Show the full solution

A square's four sides are all equal, so its perimeter is $4s$. With a side of 9 that gives $$P = 4 \times 9 = 36.$$ The perimeter is $\boxed{36}$ units.

Problem

The figure above is a rectangle with length $13$ and an unknown width. Its perimeter is $40$. Find the width. Give the number of units.

Show a hint

The perimeter $2(l+w)$ is made of two lengths and two widths, so half of it is exactly one length plus one width. Start by halving $40$.
Half the perimeter is $\frac{40}{2}=20$, and that equals one length plus one width. You already know the length is $13$, so subtract it out to uncover the width.

Show the full solution

Half the perimeter is one length plus one width, and half of $40$ is $20$. The length is $13$, so the width is $$20 - 13 = \boxed{7}.$$ Halving first is faster than writing $2(13+w)=40$ and unwinding it, though both get there.

Problem

In the diagram above is a square whose perimeter is $48$. Find the length of one side. Give the number of units.

Show a hint

A square has four equal sides, so the perimeter is $4s$. You know the perimeter and want the side.
Undo the multiplication by 4. Divide the perimeter by 4 to get one side.

Show the full solution

A square's four sides are equal, so its perimeter is $4s$. Here $4s = 48$, so divide both sides by 4 to get one side. $$s = \frac{48}{4} = \boxed{12}$$ One side is 12 units.

Problem

The diagram above shows an L-shaped figure with its six sides marked $9$, $3$, $5$, $3$, $4$, and $6$. Find its perimeter. Give the number of units.

Show a hint

The perimeter of any figure is the total distance around it, so it's just the sum of all its sides. Here every side you need is already marked.
Add the six marked lengths together, going once around the figure. $9 + 3 + 5 + 3 + 4 + 6$.

Show the full solution

Every side is marked, so add all six. $$9 + 3 + 5 + 3 + 4 + 6 = \boxed{30}$$ The perimeter is $30$ units. An L-shape has no formula, and it does not need one, since perimeter is always just the trip around.

Problem

The figure above is an L-shaped room. Its six walls measure $16$, $10$, $7$, $4$, $9$, and $6$. A builder wants to run baseboard trim all the way around the floor, along every wall. Find the total length of baseboard needed, which is the perimeter of the room. Give the number of units.

Show a hint

Baseboard runs along every wall with no gaps, so its total length is the perimeter, the sum of all six sides.
Every side is already marked on the figure above, so no side needs to be deduced. Add all six lengths together.

Show the full solution

Baseboard runs along every wall with no gaps, so its length is the perimeter. All six walls are marked, so add them. $$16 + 10 + 7 + 4 + 9 + 6 = \boxed{52}$$ The builder needs $52$ units of baseboard.

Problem

In the diagram above is a staircase figure with all eight of its sides marked. The bottom and the left side are each $6$, and the six little step sides are each $2$. Add up every side to find the perimeter. Give the number of units.

Show a hint

The perimeter is the sum of all eight sides. Read each length off the diagram and lay them out in one long sum before you add.
There is a shortcut worth noticing. A staircase that only steps outward wraps the same distance as the rectangle that boxes it in, so the three horizontal steps together span the same width as the $6$ at the bottom, and the three vertical steps span the same height as the $6$ on the left.

Show the full solution

Add all eight sides, the $6$ along the bottom, the $6$ up the left, and the six step sides of $2$. $$6 + 2 + 2 + 2 + 2 + 2 + 2 + 6 = \boxed{24}$$ The three horizontal steps span the same width as the bottom $6$, and the three vertical steps the same height as the left $6$, so the outline wraps as far as the $6$-by-$6$ box around it, $2(6+6)=24$.

Problem

In the diagram above is a staircase figure. Only its overall width $12$ along the bottom and its overall height $8$ up the left side are marked, and every other side is left blank. Find its perimeter. Give the number of units.

Show a hint

A staircase only steps outward, so slide each horizontal piece up and each vertical piece across until the outline snaps onto the smallest rectangle that would just contain it. Nothing about the total length changes.
That bounding rectangle is $12$ wide and $8$ tall, so its perimeter is $2(W+H)$. Read off $W$ and $H$ from the two marked sides.

Show the full solution

Slide every horizontal step up to the top and every vertical step out to the right. The staircase only steps outward, so nothing overlaps and no length is lost, and the outline matches the $12$ by $8$ box around it. $$P = 2(W+H) = 2(12+8) = \boxed{40}$$ The individual step sizes never matter, which is why you were not given them.

Problem

The diagram above is a rectangle 14 by 9 with a small rectangular piece cut out of one corner. Its six sides, going around, are $14$, $7$, $3$, $2$, $11$, and $9$. Find the perimeter. Give the number of units.

Show a hint

A corner notch pushes the boundary in and then straight back out. The two little sides you gained replace exactly the length you gave up, so the perimeter matches the full rectangle the piece was cut from.
That full rectangle is 14 by 9, so its perimeter is $2(14+9)$. Or add the six marked sides straight off the figure and check they agree.

Show the full solution

Add the six marked sides. $$14+7+3+2+11+9 = \boxed{46}$$ That matches $2(14+9)$, the perimeter of the full $14$ by $9$ rectangle. A corner notch steps in and back out, so its two new sides cover exactly the edge they replaced.

Problem

In the diagram above is a rectangle 12 units wide and 8 units tall, with a slot cut into its top edge 3 units wide and 2 units deep. Find the perimeter. Give the number of units.

Show a hint

Start with the plain rectangle, before the slot. Its perimeter is $2(l+w)$, so $2(12+8)$.
The slot's 3-unit width is already part of the top edge, so cutting it costs nothing there. What it adds are the two new vertical walls of the slot, each 2 units deep. Add twice the depth to the rectangle's perimeter.

Show the full solution

The full rectangle has perimeter $2(12+8) = 40$. The slot's two side walls are new, each 2 units deep, so they add $2\times 2 = 4$. $$40 + 4 = \boxed{44}$$ The slot's 3-unit width costs nothing, since the strip taken off the top edge comes back as the slot's floor. Only the depth counts, and it counts twice.

Problem

The rectangle in the diagram above has length $(x+3)$ and width $2x$, and its perimeter is $30$. Find $x$. Give the value of $x$.

Show a hint

Perimeter of a rectangle is $2(l+w)$. Add the length and width, double it, and set that equal to $30$.
Once you have $2((x+3)+2x)=30$, divide both sides by $2$ first. That leaves a simple equation in $x$ to solve.

Show the full solution

Perimeter is $2(l+w)$, so $2((x+3)+2x)=30$. Divide both sides by $2$ to get $(x+3)+2x=15$, which is $3x+3=15$. Then $3x=12$, so $x=\boxed{4}$.

Problem

In the diagram above is a triangle whose three sides are $x$, $(x+2)$, and $(x+4)$, and its perimeter is $27$. Find the LONGEST side. Give the number of units.

Show a hint

Perimeter is the sum of all three sides, so add the three labels and set that equal to $27$.
Combine the $x$ terms to get a single equation in $x$, solve it, then plug back into the longest label, $x+4$.

Show the full solution

The perimeter is the sum of the three sides, so add the labels and set the total to $27$.$$x + (x+2) + (x+4) = 27$$Combine like terms to get $3x + 6 = 27$, so $3x = 21$ and $x = 7$. The longest side is $x+4$, which is $7+4 = \boxed{11}$.

Problem

In the diagram above is a U-shaped figure with a deep slot cut down from the top. Its eight sides, going around, are $12$, $8$, $4$, $5$, $4$, $5$, $4$, and $8$. Find its perimeter. Give the number of units.

Show a hint

Perimeter is the whole trip around the outside, so you just add every side you walk along. The slot doesn't change that rule. Both walls of the slot are part of the outside, so both get counted.
Add all eight numbers on the diagram. Group them however is easiest, and make sure both $5$-unit slot walls are in your sum.

Show the full solution

Add all eight sides. $$12 + 8 + 4 + 5 + 4 + 5 + 4 + 8 = \boxed{50}$$ Both $5$-unit walls of the slot count. You walk down one and back up the other, so a slot adds two lengths to the trip, not one.

Practice these ideas

Practice

In the diagram above is a rectangle 15 units wide and 8 units tall. Find its perimeter. Give the number of units.

Show the solution

Add the width and the height, then double. $$2(15+8) = 2 \times 23 = \boxed{46}$$ The perimeter is $46$ units. The doubling works because opposite sides of a rectangle are equal.

Practice

The diagram above shows a square with each side 11 units long. Find its perimeter. Give the number of units.

Show the solution

A square is four copies of the same side, so its perimeter is $4s$. With $s = 11$, $$4 \times 11 = 44.$$ The perimeter is $\boxed{44}$ units.

Practice

The rectangle in the diagram above is 16 units long, and its perimeter is 50. Find its width. Give the number of units.

Show the solution

A rectangle's perimeter is $2(l+w)$, so half of it is exactly one length plus one width. Half of $50$ is $25$, and that equals the length plus the width. The length is $16$, so the width is $$25 - 16 = \boxed{9}.$$

Practice

The diagram above shows a square whose perimeter is $52$. Find the length of one side. Give the number of units.

Show the solution

All four sides of a square are equal, so the perimeter is $4s$. Setting that equal to $52$ and dividing by $4$ gives the side.$$s = \frac{52}{4} = \boxed{13}$$

Practice

In the diagram above is an L-shape with its six sides marked $8$, $4$, $5$, $3$, $3$, and $7$. Find its perimeter. Give the number of units.

Show the solution

Add the six marked sides. $$8+4+5+3+3+7=\boxed{30}$$ The perimeter is $30$ units. There is no formula for an L-shape, so walk once around and add each side as you pass it.

Practice

In the diagram above is a staircase figure with its six sides marked $8$, $3$, $4$, $3$, $4$, and $6$. Find its perimeter. Give the number of units.

Show the solution

Perimeter is the total distance around the figure, so add all six marked sides. $$8+3+4+3+4+6 = \boxed{28}$$ The perimeter is $28$ units.

Practice

The staircase in the diagram above steps outward from bottom to top, and only two lengths are marked: the total width $10$ along the bottom and the total height $7$ up the left side. Find its perimeter. Give the number of units.

Show the solution

A staircase that only steps outward wraps the same distance as the smallest rectangle around it. The horizontal pieces up top add to the same $10$ as the bottom, and the vertical pieces add to the same $7$ as the left side. $$P = 2(10 + 7) = \boxed{34}$$ The individual step sizes never come into it, which is why you were only given the two totals.

Practice

The figure above is a 12-by-8 rectangle with a small notch cut out of one corner. Find its perimeter. Give the number of units.

Show the solution

A notch does not change the perimeter, so use the full 12-by-8 rectangle. $$2(12+8) = \boxed{40}$$ The two short pieces the notch removes from the outer edges are exactly replaced by the two sides of the notch itself, so the trip around comes out the same.

Practice

The figure above is an 11-by-7 rectangle with a slot 2 units wide and 3 units deep cut straight into its top edge. Find the perimeter of this figure. Give the number of units.

Show the solution

The full rectangle is $2(11+7) = 36$, and the slot adds twice its depth for its two walls, $2\times 3 = 6$. $$36 + 6 = \boxed{42}$$ The slot's 2-unit width changes nothing, since the strip taken off the top edge comes back as the slot's floor.

Practice

In the diagram above is a rectangle whose length is $(x+5)$ and whose width is $3x$. Its perimeter is $42$. Find the value of $x$. Give the value of $x$.

Show the solution

The perimeter is $2(l+w)$, so one length plus one width is half of it, $21$. That gives $$(x+5)+3x = 21.$$ Combine the $x$ terms to get $4x + 5 = 21$, so $4x = 16$ and $x = \boxed{4}$.

Practice

In the diagram above is an isosceles triangle whose sides are labelled $x$, $2x$, and $2x$, and its perimeter is $40$. Find the length of the longest side. Give the number of units.

Show the solution

The perimeter is the sum of all three sides, so $x + 2x + 2x = 5x$. Setting that equal to the given perimeter gives $$5x = 40,\qquad x = 8.$$ The longest side is $2x = 2(8) = \boxed{16}$.

Practice

In the diagram above is an L-shaped patio. Its walls, going around, measure $15, 9, 6, 4, 9,$ and $5$ units. Find the perimeter of the patio. Give the number of units.

Show the solution

Perimeter is the total distance around the figure, so add all six walls in order. $$15+9+6+4+9+5 = \boxed{48}$$ The patio's perimeter is $48$ units.

Practice

The diagram above shows a U-shape with its sides marked $14, 9, 5, 6, 4, 6, 5, 9$. Find its perimeter. Give the number of units.

Show the solution

The perimeter of any figure is the sum of all its sides, so read the eight marked lengths off the diagram and add them up. $$14+9+5+6+4+6+5+9 = \boxed{58}$$ The perimeter is $58$ units.