Grade 3 · 3.MD.D.8 · 4.MD.A.3

Perimeter

A farmer is buying fence for a goat pen, a strip of wood for a rabbit hutch, baseboard for a room. How much does it take to go all the way around a flat shape, and how do you find it when some sides are not marked?

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Measuring lengthGrade 1

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Finding a missing side from the perimeterGrade 3 Perimeter versus areaGrade 3

Where you are 15 in Area and perimeter

Measurement12

Area and perimeter15

▴ 2 above Splitting a rectangle into rows and columns of squares Area of rectangles and squares

Perimeter

Area of parallelograms and trapezoids Finding a missing side from the area ▾ Show all 15 (8 more)

When you reach for it

Edurne is putting up a fence around a goat pen. The fence goes along every edge of the pen, all the way around, and ends where it started. How much fence she needs is a length: the distance all the way around the pen. That distance is the pen’s perimeter.

The perimeter of a flat shape is the total length of its boundary, the distance you would walk if you went once around its edge and came back to where you began.

Because the perimeter is a length, it is measured in units of length: centimeters, meters, inches, feet. You measure it the way you measure any length (see Measuring length: lay a unit end to end and count, or read a ruler), only the thing being measured bends at the corners. Fence, a wood trim, a ribbon around a card, baseboard along the walls of a room: each of these is a perimeter question.

Perimeter is not the same as area. In a picture of a field with a fence around it, the perimeter is the fence and the area is the field. Area counts the squares that cover the inside (Area as covering); perimeter measures only the edge, in plain length units.

The lesson takes three days. Part 1 finds the perimeter of any shape with straight sides by measuring or adding its sides. Part 2 does rectangles and squares, where equal sides give shortcuts. Part 3 handles shapes with square corners where some sides are not marked and have to be worked out from the others first.

The procedure

Part 1: all the way around

A shape with straight sides, a polygon, has a perimeter equal to the sum of its side lengths. That is just what “all the way around” means: the walk around the edge is made of the sides, one after another, each walked once.

So the work is to know every side, in the same unit, and add. A side can be known in three ways. It can be marked on a drawing. It can be measured with a ruler or a tape. Or, on a grid of unit squares, it can be counted: each side of a small square is one unit of length, so a side that runs along 4 squares is 4 units long.

Two cautions keep the sum honest. Every side must be counted, and no side counted twice; going around in order, starting at one corner and stopping when you return to it, takes care of both. And only the outside edge counts. On a grid, the lines inside the shape separate squares, but a fence would never run along them, so they are not part of the perimeter.

The sides can be added in any order and grouped in any way. Changing the order is the Properties of addition, and changing the grouping is the associative property of addition. Both let you pick friendly pairs: in 5 + 4 + 5, adding the two 5s first gives 10 + 4 = 14.

Part 2: rectangles and squares

A rectangle has four sides and four square corners, and its opposite sides are equal (see Rectangles, rhombuses and squares). The two long sides are the length and the two short sides the width. So a rectangle’s perimeter needs only two numbers. Going around, the sides are length, width, length, width:

P = \text{length} + \text{width} + \text{length} + \text{width}

Grouping the two lengths together and the two widths together gives a second way to write it, and grouping each length with the width next to it gives a third:

\begin{aligned} P &= 2 \times \text{length} + 2 \times \text{width} \\ P &= (\text{length} + \text{width}) \times 2 \end{aligned}

All three give the same number, because they add the same four sides. The third reads as: walk one length and one width, which takes you halfway around, then do it again.

A square is a rectangle whose four sides are all equal. Its perimeter is one side added four times, which is 4 \times \text{side}.

Part 3: sides you have to work out

Many real shapes have square corners but are not rectangles: an L-shaped room, a pen with steps cut into one corner. A drawing of such a shape often marks only some of its sides. The others are not lost. With square corners, every side runs either across or up and down, and that fixes the missing sides.

Look at the L-shaped room in Example 5. Its bottom wall runs across the whole room. The top of the room is in two pieces at different heights, but they still run across the whole room between them, so together they are as long as the bottom wall. The same goes for up-and-down sides: the pieces on the right add up to the whole left side. So, for an L or a staircase whose steps all go one way,

the across pieces on top together equal the bottom, and
the up-and-down pieces on one side together equal the side opposite.

A missing piece is the whole minus the pieces you know, which is a subtraction. Once every side is known, add them as in Part 1. Checking that each side has a number before adding is what keeps a side from being skipped.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.

Example 1

A five-sided chicken run

Mikel builds a chicken run with five straight sides, measuring 5 meters, 5 meters, 4 meters, 7 meters and 3 meters. How many meters of fence go around it?

Figure 1Five sides, each one measured. The fence has to cover every one of them once.

Solution

The fence goes along every side once, so its length is the sum of the five sides:

5 + 5 + 4 + 7 + 3

The order and grouping can be changed (the commutative and associative properties), so pick pairs that make ten: 5 + 5 = 10 and 7 + 3 = 10. That leaves 10 + 10 + 4 = 24.

Mikel needs 24 meters of fence. The answer is in meters because every side was in meters.

Example 2

Counting around a shape on a grid

Joseba draws a T shape on centimeter grid paper: a bar of 3 squares across the top, and a stem of 2 squares hanging below the middle square. What is its perimeter?

Figure 2Every square is 1 centimeter on a side. Only the dark outline counts toward the perimeter; the dashed lines inside do not.

Solution

Each side of a small square is 1 centimeter, so the perimeter is the number of square sides along the dark outline. Go around in order. Across the top of the bar: 3. Down the right end of the bar: 1. Back along the underside of the bar, right of the stem: 1. Down the right side of the stem: 2. Across the bottom of the stem: 1. Up the left side of the stem: 2. Along the underside of the bar, left of the stem: 1. Up the left end of the bar: 1. The sum is

3 + 1 + 1 + 2 + 1 + 2 + 1 + 1 = 12

The perimeter is 12 centimeters. The dashed lines between the squares inside the T are not counted: they are not on the edge.

Example 3

Fence for the goat pen

Edurne’s goat pen is a rectangle 9 meters long and 6 meters wide. How much fence goes around it?

Figure 3A rectangle’s opposite sides are equal, so two numbers tell all four sides.

Solution

The opposite sides of a rectangle are equal, so the four sides are 9, 6, 9 and 6 meters. Going around:

9 + 6 + 9 + 6 = 30

The two other ways give the same total. Two lengths and two widths: 2 \times 9 + 2 \times 6 = 18 + 12 = 30. One length and one width is halfway around, 9 + 6 = 15, and doubling it gives 15 + 15 = 30.

Edurne needs 30 meters of fence.

Example 4

Trim for a square rabbit hutch

The floor of Joseba’s rabbit hutch is a square with 4-foot sides. He nails a strip of wood along every edge of the floor. How long is the strip?

Solution

All four sides of a square are equal, so the strip is 4 + 4 + 4 + 4 feet, which is 4 \times 4 = 16 feet.

The strip is 16 feet long. The floor also happens to cover 16 square feet, but that is a different quantity, an area in square units; how the two numbers relate is the subject of Perimeter versus area.

Example 5

Baseboard around an L-shaped room

Mikel is putting baseboard along every wall of an L-shaped room. The plan marks the bottom wall 16 feet, the short right wall 5 feet, the top wall 10 feet and the left wall 12 feet. The two walls at the inside corner are not marked. How much baseboard does he need, if it runs along every wall?

Figure 4Two walls are not marked, but the walls that are marked decide them.

Solution

First find the two missing walls. The bottom wall runs across the whole room, 16 feet. Across the top, the room has the 10-foot wall and the unmarked wall at the inside corner, and together they span the same width, so the unmarked across wall is

16 - 10 = 6 \text{ feet}

Up and down, the left wall is 12 feet. On the right side the room has the 5-foot wall and the unmarked up-and-down wall, which together span the same height, so that wall is

12 - 5 = 7 \text{ feet}

Now every wall is known. Going around from the bottom: 16, 5, 6, 7, 10 and 12 feet. Adding,

16 + 5 + 6 + 7 + 10 + 12 = 56

Mikel needs 56 feet of baseboard. Check: the six walls make the trip all the way around, and each has been counted once.

Example 6

A stepped chicken run

Edurne fences a run with three steps cut along one side. The bottom is 9 meters and the left side 6 meters. From the bottom right corner, the edge goes up 2 meters, across 3 meters, up 2 meters, across 3 meters, then up and across to the top left corner by two unmarked sides. Find the perimeter.

Figure 5The across pieces of the steps must add up to the bottom, and the up pieces to the left side.

Solution

The across pieces of the steps, including the unmarked top, run across the whole run between them, so they add up to the bottom, 9 meters. Two of them are 3 meters, so the top is 9 - 3 - 3 = 3 meters.

The up pieces on the right add up to the left side, 6 meters. Two of them are 2 meters, so the last one is 6 - 2 - 2 = 2 meters.

Going around from the bottom: 9, 2, 3, 2, 3, 2, 3 and 6 meters. Group the across pieces and the up pieces:

\begin{aligned} &(9 + 3 + 3 + 3) + (6 + 2 + 2 + 2) \\ &= 18 + 12 = 30 \end{aligned}

The perimeter is 30 meters.

Example 7

The same fence as the goat pen

The stepped run of Example 6 fits inside a 9 by 6 rectangle, the size of the goat pen in Example 3. Both need 30 meters of fence. Is that an accident? Explain why the stepped run’s perimeter had to come out the same.

Figure 6Slide the steps out and they fill the dashed top and right side exactly. The stepped run needs the same fence as the 9 by 6 goat pen.

Solution

In Example 6, the three across pieces of the steps added up to 9, the same as the top of the rectangle. Slide each across piece straight up and the three of them lie end to end along the dashed top, covering it exactly. In the same way, the three up pieces added up to 6; slide each one straight right and they cover the dashed right side exactly.

So the stepped run’s edge is the rectangle’s edge cut into pieces and moved: the bottom and left side are shared, the top is the across pieces, the right side is the up pieces. Same pieces, same total length, so the same perimeter, 2 \times 9 + 2 \times 6 = 30 meters. It is not an accident.

This gives a shortcut for any shape whose corners are all square and whose steps all go one way, like an L or a staircase: its perimeter equals that of the rectangle drawn snugly around it. The shortcut fails for a shape with a notch cut into a side, as the Boundaries explain.

Boundaries

The shape must be closed. A perimeter is a trip that ends where it began. A path with two loose ends, such as a fence along only three sides of a pen, has a length but not a perimeter.

All sides in the same unit. Adding 3 meters to 40 centimeters does not give 43 of anything. Change one length into the other’s unit first (Metric length units, Customary length units), then add.

Only the outside edge. Lines inside a shape, grid lines or a wall splitting one pen into two, are not part of its perimeter. A fence for the two pens together is a different question from the perimeter of the outside.

The rectangle shortcuts need a rectangle. 2 \times \text{length} + 2 \times \text{width} works because opposite sides are equal. For a shape with four sides that are not two equal pairs, add all four.

Working out a missing side needs square corners. The rule “the pieces on top add up to the bottom” holds because every side runs across or up and down. On a shape with slanted sides, a missing side cannot be found that way; it has to be measured or marked.

The rectangle-around-it shortcut of Example 7 needs steps that go one way. If a notch is cut into a side, as in a U shape, the two inside walls of the notch are extra edge with nothing to slide onto, and the perimeter is more than that of the rectangle around it (Exercise 19).

Curved edges cannot be added side by side. A round pen has no straight sides. Its perimeter is still a length, found by laying a string or a tape along the curve and measuring it.

Where this goes next

Working forward, this lesson adds known sides to get the perimeter. Finding a missing side from the perimeter runs it backward: the perimeter is known, along with all but one side, and the last side is found. Perimeter versus area sets the fence and the field side by side and shows that two shapes with the same perimeter can cover very different areas. For an L-shaped room’s area, rather than its edge, the shape is split into rectangles in Splitting a figure into rectangles.

Much later, perimeter turns up inside other formulas. The distance around a circle has its own name, the circumference. And one formula finds a triangle’s area from its three sides using half its perimeter (Heron's formula).

Exercises

Part 1: all the way around

A triangle-shaped sign has sides of 6 inches, 8 inches and 9 inches. What is its perimeter?
A garden bed has four sides: 12 feet, 9 feet, 15 feet and 10 feet. What is its perimeter?
A tile has six sides, each 5 centimeters long. What is its perimeter?
A chicken run has six sides: 4 meters, 6 meters, 3 meters, 5 meters, 2 meters and 8 meters. How much fence goes around it? Group the sides to make the adding easy.
Find the perimeter of shape A in the figure.
Find the perimeter of shape B in the figure.
Find the perimeter of shape C in the figure.

Figure 7Shapes for Exercises 5 to 7. Each small square is 1 centimeter on a side.

A field has four sides of 120 meters, 85 meters, 140 meters and 95 meters. How far is it all the way around?

Part 2: rectangles and squares

A chicken coop is a rectangle 7 meters long and 4 meters wide. Find its perimeter.
The floor of a coop is a square, 8 feet on each side. A board runs along every edge. How long is the board, all together?
A rabbit run is a rectangle 10 feet long and 3 feet wide. How much fencing goes around it?
A hutch door is a rectangle 25 inches tall and 15 inches wide. A weather strip goes around its edge. How long is the strip?
One goat pen is a square with 9-meter sides. Another is a rectangle 12 meters long and 5 meters wide. Which needs more fence, and how much more?
A pasture is a rectangle 40 meters long and 30 meters wide. Find its perimeter using length plus width, doubled.
Two rabbit runs are each a rectangle 6 feet long and 2 feet wide. Edurne pushes them together, long side against long side, to make one run 6 feet long and 4 feet wide. What is the perimeter of one small run, and of the big run? Why is the big run’s perimeter not twice a small one’s?

Part 3: sides you have to work out

Find the two missing sides of shape P, then its perimeter.
Shape Q is the floor plan of an L-shaped room. Find the two missing walls and the perimeter. Baseboard goes along every wall except a doorway 3 feet wide in the bottom wall. How much baseboard is needed?
Find the two missing sides of the staircase R, then its perimeter.
Find the two missing sides of the U shape S, then its perimeter.

Figure 8Shapes for Exercises 16 to 19. Every corner is a square corner.

Mikel’s rabbit run starts as a rectangle 6 meters long and 4 meters wide. He leaves out a corner piece 2 meters long and 1 meter wide, so the run has six sides, all meeting at square corners. What is the perimeter of the run?

Explain your thinking

The perimeter of the goat pen in Example 3 is 30 meters, not 30 square meters. Explain the difference, using the fence and the field.
A rectangle has sides that are whole numbers of meters. Can its perimeter be an odd number of meters? Explain.
Explain why adding the length and the width of a rectangle and then doubling the sum always gives its perimeter.
An L-shaped pen has square corners. Its bottom side is 12 feet and its left side 7 feet, and none of the other four sides is marked. Can you still find its perimeter? If so, find it and explain.
The U shape S in Exercise 19 fits snugly inside a rectangle 9 meters by 5 meters. Find that rectangle’s perimeter, and explain why S needs more fence than the rectangle does.

Answers

23 inches. 6 + 8 + 9 = 14 + 9 = 23.
46 feet. 12 + 9 = 21 and 15 + 10 = 25, then 21 + 25 = 46.
30 centimeters. Six equal sides of 5: 6 \times 5 = 30.
28 meters. Group as (4 + 6) + (3 + 5 + 2) + 8 = 10 + 10 + 8 = 28.
8 centimeters. Around shape A, the outline is 8 square sides long: across the top 1, down 1, across 1, down 1, along the bottom 2, up the left side 2, and 1 + 1 + 1 + 1 + 2 + 2 = 8.
10 centimeters. Top 3, down the right end 1, back along the underside 1, down 1, bottom 2, up the left side 2: 3 + 1 + 1 + 1 + 2 + 2 = 10.
12 centimeters. The left side is 3 and the bottom is 3. The steps go up 1, across 1, up 1, across 1, up 1, across 1, so they add 6 more: 3 + 3 + 6 = 12.
440 meters. 120 + 85 = 205, 205 + 140 = 345, 345 + 95 = 440.
22 meters. 7 + 4 = 11 is halfway around, and 11 + 11 = 22.
32 feet. 4 \times 8 = 32.
26 feet. 2 \times 10 + 2 \times 3 = 20 + 6 = 26.
80 inches. 25 + 15 = 40, doubled, 40 \times 2 = 80.
The square pen, by 2 meters. The square needs 4 \times 9 = 36 meters; the rectangle needs 2 \times 12 + 2 \times 5 = 24 + 10 = 34 meters; 36 - 34 = 2.
140 meters. 40 + 30 = 70, and 70 \times 2 = 140.
A small run is 2 \times 6 + 2 \times 2 = 16 feet; the big run is 2 \times 6 + 2 \times 4 = 20 feet. Twice 16 is 32, but the two long sides that were pushed together are now inside the big run, not on its edge: 32 - 6 - 6 = 20.
The missing across side is 10 - 4 = 6 meters and the missing up-and-down side is 8 - 3 = 5 meters. Perimeter 10 + 3 + 6 + 5 + 4 + 8 = 36 meters.
The missing walls are 14 - 9 = 5 feet and 11 - 6 = 5 feet. Perimeter 14 + 6 + 5 + 5 + 9 + 11 = 50 feet. Leaving out the doorway, the baseboard is 50 - 3 = 47 feet.
The top is 8 - 3 - 3 = 2 meters and the last up side is 9 - 3 - 2 = 4 meters. Perimeter 8 + 3 + 3 + 2 + 3 + 4 + 2 + 9 = 34 meters.
The notch’s bottom is 9 - 3 - 3 = 3 meters. Both top pieces are at the same height, so the notch’s left side is as deep as its right side, 2 meters. Perimeter 9 + 5 + 3 + 2 + 3 + 2 + 3 + 5 = 32 meters.
20 meters. With the long side 6 along the bottom and the corner left out at the top right, the sides are 6 (bottom), 4 - 1 = 3 (right), 2 (across the cut), 1 (up the cut), 6 - 2 = 4 (top) and 4 (left): 6 + 3 + 2 + 1 + 4 + 4 = 20. That equals the full rectangle’s 2 \times 6 + 2 \times 4 = 20, as Example 7 predicts.
Perimeter is a length: 30 meters of fence laid out straight would reach 30 meters. Square meters measure the field inside, counted in squares 1 meter on a side. The fence goes around the field; it does not cover it.
No. The perimeter is (\text{length} + \text{width}) \times 2, and any whole number times 2 is even.
Going around a rectangle, the sides come in the order length, width, length, width. The first length and width take you halfway around, to the opposite corner, and the second length and width bring you back. Because opposite sides are equal, the two halves are the same, so the perimeter is the length plus the width, twice.
Yes: 38 feet. With square corners and steps that go one way, the across pieces on top add up to the bottom, 12, and the up-and-down pieces on the right add up to the left side, 7. So the perimeter is the same as a 12 by 7 rectangle’s: 2 \times 12 + 2 \times 7 = 24 + 14 = 38.
The rectangle’s perimeter is 2 \times 9 + 2 \times 5 = 18 + 10 = 28 meters, and S’s is 32 meters. The top pieces of S (3, 3 and the notch’s bottom, 3) slide up to cover the rectangle’s top, but the two inside walls of the notch, 2 + 2 = 4 meters, have nowhere to slide: they are extra edge. 28 + 4 = 32.