Perimeter versus area
A sheep pen needs fence around it and grass inside it. Does the amount of fence decide the amount of grass, and if the fence is fixed, which pen gives the sheep the most room?
Where you are 15 in Area and perimeter
Why this exists
Zivile keeps sheep. Their pen needs two things: a fence that goes all the way around it, and grass that covers the ground inside it. Fence is bought by the meter. Grass is laid in squares. Both depend on how big the pen is, so it is easy to think that one number tells both.
It does not. Two pens can use exactly the same fence and hold very different amounts of grass. Two pens can hold exactly the same grass and need very different amounts of fence. The edge and the inside are two separate measures, and a question about a shape has to say which one it wants.
Perimeter measured the edge, and Finding a missing side from the perimeter worked it backward. Area as covering and Area of rectangles and squares counted the squares inside. This lesson sets the two side by side. It takes three days. Part 1 tells the two measures apart and decides which one a question needs. Part 2 lists every rectangle with a given perimeter, or a given area, in an organized table, and sees how much the other measure changes. Part 3 reads those tables for the best choice: the most room for a fixed fence, and the least edge for a fixed set of squares.
Definition
The perimeter of a flat shape is the length of its edge, all the way around. It is a length, so it is measured in units of length: meters, feet, inches, centimeters.
The area of a flat shape is the amount of flat space inside it: the number of unit squares that cover it with no gaps and no overlaps. It is counted in square units. A square meter is a square 1 meter on each side; a square foot is a square 1 foot on each side.
For a rectangle with whole-number sides, the two measures are found from the same two numbers in two different ways. Going around adds the four sides; covering counts rows of squares:
The unit says which measure a number is. “20 feet” is a perimeter or some other length. “24 square feet” is an area.
Intuition
Part 1: the fence and the field
On grid paper the difference can be seen. In the playpen below, each small square is 1 foot on a side. The perimeter is the number of square sides along the dark outline. The area is the number of squares. One side of a square is 1 foot, a length you could lay a ruler along; one whole square is 1 square foot, a patch you could put your hand on.
To decide which measure a question needs, ask what the stuff in the question does. Fence, a border, ribbon, trim and tape run along the edge: they are laid out in a line, so they are measured by length, and the question wants the perimeter. Carpet, grass, paint, tiles and floor mats spread over the inside: they are laid down in patches, so they are counted in squares, and the question wants the area. Words like “around” and “edge”, or “cover” and “inside”, often point the way, but the reason is what the material does.
Part 2: same fence, different room; same room, different fence
Hold the perimeter still. Going around a rectangle walks length, width, length, width, so one length and one width use half the fence (Finding a missing side from the perimeter). With 12 meters of fence, every rectangle has length plus width equal to 6. There are several such rectangles, and they do not cover the same ground.
Hold the area still. With 12 carpet squares laid as a rectangle, the rows times the squares in each row must make 12. Several rectangles do that too, and their outlines are not the same length.
To be sure no rectangle is missed, make an organized list. Start with a width of 1, then 2, then 3, and so on, and work out the length that goes with each. Stop when the width would be longer than the length, because from then on every rectangle is one already listed, turned on its side. Going in order is what makes the list complete: each width has been tried exactly once.
Part 3: closest to a square
Read down a list with a fixed perimeter and the areas grow as the length and the width come closer together. The biggest area is at the bottom, where the two sides are as close as whole numbers allow: a square when the sides can be equal, and sides that differ by 1 when they cannot.
Read down a list with a fixed area and the perimeters shrink in the same way. The long thin strip, one square wide, has the most edge. The rectangle closest to a square has the least.
Results
Perimeter and area are different measures. Perimeter is a length, in units such as meters or feet. Area is a number of unit squares, in square units such as square meters or square feet. They cannot be added to each other or compared with each other.
The same perimeter can enclose different areas. With 12 meters of fence and whole-meter sides, the pens are 5 by 1, 4 by 2 and 3 by 3, with areas 5, 8 and 9 square meters.
The same area can have different perimeters. Twelve carpet squares make rectangles 12 by 1, 6 by 2 and 4 by 3, with perimeters 26, 16 and 14 feet.
Closest to a square wins. In every list of rectangles with whole-number sides in this lesson, the rectangle whose length and width are closest together has the largest area for its perimeter, and the smallest perimeter for its area. For a fixed area, the strip one unit wide has the largest perimeter.
Why it is true
Moving squares keeps the area but not the edge. Take the 12 carpet squares from a 12 by 1 strip and relay them as 4 by 3. Nothing was added or taken away, so the count of squares, the area, is still 12.
The outline is another matter. Each square has 4 edges, so 12 squares have 4 \times 12 = 48 edges in all. Wherever two squares touch, the 2 edges that meet there are inside the shape, not on its outline. In the strip, squares touch in 11 places, which hides 2 \times 11 = 22 edges and leaves 48 - 22 = 26 on the outline. In the 4 by 3 rectangle, each of the 3 rows has 3 places where squares touch, and each of the 4 columns has 2, which makes 9 + 8 = 17 places. That hides 34 edges and leaves 48 - 34 = 14. A squarer rectangle packs its squares more tightly, more of their edges face each other, and less edge is left outside.
Trading length for width, with the fence fixed. Start with the 5 by 1 sheep pen and 12 meters of fence. Shorten it by 1 meter and widen it by 1 meter: the fence is the same, since one length went down by 1 and one width went up by 1. In squares, the 1 square at the end is taken away, and a new row of 4 squares is added along the remaining length of 4. The pen gains 4 and loses 1, going from 5 to 8 square meters. Do it again, from 4 by 2 to 3 by 3: the end column of 2 squares goes, and a row of 3 comes, so the area goes from 8 to 9. One more trade would turn the 3 by 3 square into a pen 2 meters long and 4 meters wide, which is the 4 by 2 pen turned, and the area goes back down to 8.
Each trade gains a row as long as the new length and loses a column as tall as the old width. While the length is well ahead of the width, the gain is bigger, so the area keeps growing until the two sides meet in the middle. That is what every table in this lesson shows. It is a reason for these lists of whole numbers, not a proof about every rectangle; the general fact, with any lengths at all, comes much later.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 to 5 with Part 2, and Examples 6 and 7 with Part 3.
The playpen: panels and floor
Dovydas builds a playpen 6 feet long and 4 feet wide out of fence panels, each panel 2 feet long. He covers the floor with foam squares, each 1 foot on a side. How many feet of panels go around the playpen, and how many panels is that? How many foam squares cover the floor?
SolutionThe panels run along the edge, so the first question asks for the perimeter. Going around, the sides are 6, 4, 6 and 4 feet:
Each panel covers 2 feet of that, so the number of panels is 20 \div 2 = 10, because 10 \times 2 = 20. Check side by side: a 6-foot side takes 3 panels and a 4-foot side takes 2, and 3 + 2 + 3 + 2 = 10.
The foam squares cover the floor, so the second question asks for the area. The floor is 4 rows of 6 squares (Figure 1):
He needs 24 foam squares, and the floor is 24 square feet. The 20 and the 24 measure different things: 20 feet is a length of panel, 24 square feet is a patch of floor.
Which measure does each question need?
Zivile’s sheep pen is a rectangle 7 meters long and 4 meters wide.
(a) How many meters of fence go around it?
(b) How many straw mats, each a square 1 meter on a side, cover its floor?
(c) A bag of grass seed covers 4 square meters. How many bags does the floor need?
Solution(a) Fence runs along the edge, so this is the perimeter. One length and one width are halfway around, 7 + 4 = 11 meters, and the whole way is 11 + 11 = 22 meters of fence.
(b) The mats spread over the floor, so this is the area. The floor is 4 rows of 7 mats, and 4 \times 7 = 28 mats, which is 28 square meters.
(c) Seed is spread over the ground, so this is the area again: 28 square meters. Each bag covers 4 of them, so the floor needs 28 \div 4 = 7 bags, because 7 \times 4 = 28.
In (a) and (b) the material told which measure to use. In (c) the words “square meters” said it as well.
Twelve meters of fence
Vytas has 12 meters of fence panels for a rectangular sheep pen. Every side must be a whole number of meters, and he uses all the fence. List every pen he could make and the area of each. Which gives the sheep the most room?
SolutionThe four sides, length, width, length, width, use all 12 meters, so one length and one width use half, 12 \div 2 = 6 meters:
Make an organized list, trying widths 1, 2, 3 in order:
| Width | Length | Fence | Area |
|---|---|---|---|
| 1 m | 5 m | 1 + 5 + 1 + 5 = 12 m | 1 \times 5 = 5 square meters |
| 2 m | 4 m | 2 + 4 + 2 + 4 = 12 m | 2 \times 4 = 8 square meters |
| 3 m | 3 m | 3 + 3 + 3 + 3 = 12 m | 3 \times 3 = 9 square meters |
A width of 4 would give a length of 2, which is the 4 by 2 pen turned on its side, so the list stops at 3. There are exactly three pens.
Every pen uses the same 12 meters of fence, and the room inside is 5, 8 or 9 square meters. The most room comes from the square pen, 3 meters by 3 meters, with 9 square meters, almost twice as much as the 5 by 1 pen.
Twelve carpet squares
Zivile has 12 carpet squares, each 1 foot on a side. She lays all of them as one rectangle, with no gaps, and puts tape along its edge. List every rectangle she could make and the length of tape each one needs.
SolutionAll 12 squares are used, so every rectangle has an area of 12 square feet: the number of rows times the squares in each row is 12. Try widths in order, keeping only the ones that share out the 12 squares evenly:
- Width 1: one row of 12, since 1 \times 12 = 12.
- Width 2: two rows of 6, since 2 \times 6 = 12.
- Width 3: three rows of 4, since 3 \times 4 = 12.
- Width 4: four rows of 3, the 4 by 3 rectangle turned, so the list stops.
Now find each perimeter:
| Rectangle | Area | Tape around the edge |
|---|---|---|
| 12 by 1 | 12 square feet | 12 + 1 + 12 + 1 = 26 feet |
| 6 by 2 | 12 square feet | 6 + 2 + 6 + 2 = 16 feet |
| 4 by 3 | 12 square feet | 4 + 3 + 4 + 3 = 14 feet |
The area is the same each time, but the tape is 26, 16 or 14 feet. The long strip needs almost twice as much tape as the 4 by 3 rectangle.
Less fence and more grass
Vytas has two plans for a sheep pen. Plan A is 9 meters by 2 meters. Plan B is 5 meters by 4 meters. Which plan needs less fence? Which gives more grass?
SolutionFence goes around, so compare perimeters:
Grass covers the inside, so compare areas: plan A is 2 \times 9 = 18 square meters and plan B is 4 \times 5 = 20 square meters.
Plan B needs 22 - 18 = 4 fewer meters of fence and still gives 20 - 18 = 2 more square meters of grass. A longer fence does not mean more room inside. Plan A is bigger around and plan B is bigger inside, so “which pen is bigger?” has no answer until it says which measure it means.
Fourteen playpen panels
Dovydas has 14 fence panels, each 2 feet long, for a rectangular playpen. Each side is made of whole panels, and he uses all 14. Which playpen gives the most floor?
SolutionCount the sides in panels first. Going around uses all 14 panels, so one length and one width use half of them, 14 \div 2 = 7 panels. Try widths of 1, 2 and 3 panels; a width of 4 panels would give a length of 3, a playpen already listed. Each panel is 2 feet, so a side of 3 panels is 3 \times 2 = 6 feet.
| Width | Length | In feet | Floor |
|---|---|---|---|
| 1 panel | 6 panels | 2 by 12 | 2 \times 12 = 24 square feet |
| 2 panels | 5 panels | 4 by 10 | 4 \times 10 = 40 square feet |
| 3 panels | 4 panels | 6 by 8 | 6 \times 8 = 48 square feet |
The most floor is the playpen 8 feet long and 6 feet wide, with 48 square feet. Check the edge: 8 + 6 + 8 + 6 = 28 feet, which is 14 panels of 2 feet.
No square playpen is possible here. A square needs length and width equal, two equal whole numbers of panels adding to 7, and 7 is odd: 3 + 3 = 6 and 4 + 4 = 8. So the winner is the closest the two sides can come, 4 panels and 3 panels, which differ by 1.
The least tape around 36 carpet squares
Vytas lays 36 carpet squares, each 1 foot on a side, as one rectangle. He wants as little edging tape around it as possible. Which rectangle should he make, and how much tape does it need? Which rectangle would need the most?
SolutionThe area is fixed at 36 square feet. List the widths that share out 36 squares evenly, and the perimeter of each:
| Rectangle | Tape around the edge |
|---|---|
| 36 by 1 | 36 + 1 + 36 + 1 = 74 feet |
| 18 by 2 | 18 + 2 + 18 + 2 = 40 feet |
| 12 by 3 | 12 + 3 + 12 + 3 = 30 feet |
| 9 by 4 | 9 + 4 + 9 + 4 = 26 feet |
| 6 by 6 | 6 + 6 + 6 + 6 = 24 feet |
A width of 5 does not work, because 5 \times 7 = 35 and 5 \times 8 = 40 skip over 36. A width of 6 gives a length of 6, and any wider would repeat a rectangle already listed, so the list is complete.
The least tape is 24 feet, for the 6 by 6 square. The most is 74 feet, for the strip one square wide, more than three times as much for the same 36 squares.
Boundaries
A perimeter and an area cannot be added or compared. One is a length and the other a number of squares. A square 4 meters on a side has a perimeter of 16 meters and an area of 16 square meters; the numbers match, but 16 meters of fence is not 16 square meters of grass, any more than 16 minutes is 16 dollars.
The lists are for whole-number sides. The fact that the rectangle closest to a square wins was found by listing whole-number sides, and that is all it is claimed for here. With halves allowed, more rectangles join a list. With 26 meters of fence and halves allowed, a square 6 and a half meters on each side fits, which the whole-number list in Exercise 23 cannot reach.
A width of zero is not a pen. With 12 meters of fence, 6 by 0 adds to 6, but the two long sides lie on top of each other and nothing is inside. Every list starts at a width of 1.
The fence must go all the way around. If one side of a pen is a barn wall, the fence covers only three sides, and the fence used is no longer the perimeter. The lists above do not answer that question.
Rectangles only. The lists compare rectangles with each other. A pen of some other shape, with slanted sides or a curved fence, is not in any of them, and it can hold a different amount of room for the same fence.
Where this goes next
Area can also be run backward: Finding a missing side from the area starts from the area and one side and finds the other, and Area of composite figures and shaded regions finds the area of shapes made of several rectangles. The question of Part 3, the most room for a fixed amount of fence, returns much later with any lengths at all, not only whole numbers. There the area of a rectangle with a fixed perimeter becomes a formula whose largest value can be found exactly, and the square still wins (Maximum, minimum and projectile problems). The same tension between the outside and the inside shows up for solids, where a box’s wrapping and the space it holds are surface area and volume.
Exercises
Part 1: the fence and the field
- Say whether each question needs the perimeter or the area, and why: (a) the fence around a sheep pen, (b) the carpet squares that cover a floor, (c) a ribbon glued along the edge of a card, (d) the paint that covers a tabletop.
- Zivile lays carpet squares, each 1 foot on a side, as a rectangle 7 squares long and 2 squares wide. How many squares does she use? How many feet of tape go around its edge?
- A sheep pen is 8 meters long and 5 meters wide. How much fence goes around it? How much ground does it cover?
- Dovydas builds a square playpen with sides of 6 feet out of panels that are each 2 feet long. How many panels does he need? How many square feet of floor are inside?
- Vytas writes that a rectangular sheep pen needs 34 ___ of fence and has 72 ___ of grass. Fill in each unit, meters or square meters. Then find a length and a width, in whole meters, that fit both numbers.
- A sheep pen is 9 meters long and 4 meters wide. One bag of grass seed covers 9 square meters. How many bags does the floor need? How much fence goes around the pen?
Part 2: organized lists
- List every rectangle with whole-meter sides that 16 meters of fence goes around exactly, and give the area of each.
- Zivile lays 18 carpet squares, each 1 foot on a side, as a rectangle. List every rectangle she can make and the perimeter of each.
- List every rectangle that 24 carpet squares, each 1 foot on a side, can make, and the perimeter of each.
- Pen A is 8 meters by 4 meters and pen B is 16 meters by 2 meters. Show that they hold the same amount of grass. Which needs more fence, and how much more?
- One pen is 10 meters by 2 meters and another is 7 meters by 5 meters. Show that they need the same length of fence. Which gives more room, and how much more?
- Find every rectangle with whole-number sides, in centimeters, whose area is 16 square centimeters, and give the perimeter of each.
Part 3: the best choice
- Vytas has 20 meters of fence for a rectangular sheep pen with whole-meter sides. List every pen, and find the one with the most room.
- With 22 meters of fence and whole-meter sides, which rectangular pen gives the most room, and how much?
- Dovydas has 18 panels, each 2 feet long, for a rectangular playpen whose sides are whole panels. Which playpen gives the most floor? Give its sides in feet and its area.
- Zivile lays 30 carpet squares, each 1 foot on a side, as a rectangle, using as little tape around the edge as possible. Which rectangle, and how much tape?
- Vytas has 28 meters of fence. Which rectangular pen with whole-meter sides gives the sheep the most room, and how much?
- Dovydas lays 40 carpet squares, each 1 foot on a side, as a rectangle and wants as much edge as possible for a ribbon border. Which rectangle gives the most edge, and how long is it?
- A playpen is 10 feet long and 6 feet wide. (a) How many panels, each 2 feet long, go around it? (b) How many foam squares, each 1 foot on a side, cover its floor? (c) A can of floor polish covers 10 square feet. How many cans does the floor need?
Explain your thinking
- Is it true that of two rectangles, the one with the longer perimeter always has the larger area? Give an example.
- Find a rectangle with whole-number sides that needs less fence than a pen 9 meters by 1 meter but holds more grass.
- In Example 3, the list of pens for 12 meters of fence starts at a width of 1. Explain why a pen 6 meters by 0 meters is not in the list.
- With 26 meters of fence and whole-meter sides, explain why no square pen is possible, and find the pen with the most room.
- Zivile takes the 18 carpet squares of an 18 by 1 strip and lays them again as a 6 by 3 rectangle. By counting the square edges hidden where two squares touch, explain why the area stays the same but the perimeter changes.
Answers
- (a) Perimeter: the fence runs along the edge. (b) Area: the squares spread over the floor. (c) Perimeter: the ribbon runs along the edge. (d) Area: the paint spreads over the top.
- 14 squares and 18 feet of tape. 2 \times 7 = 14, and 7 + 2 + 7 + 2 = 18.
- 26 meters of fence and 40 square meters of ground. Halfway around is 8 + 5 = 13, and 13 + 13 = 26; the ground is 5 \times 8 = 40.
- 12 panels and 36 square feet. The edge is 4 \times 6 = 24 feet, and 24 \div 2 = 12 panels; the floor is 6 \times 6 = 36 square feet.
- 34 meters of fence (a length) and 72 square meters of grass (a covering). Length plus width is 34 \div 2 = 17, and 9 \times 8 = 72, so the pen is 9 meters by 8 meters. Check: 9 + 8 + 9 + 8 = 34.
- 4 bags and 26 meters of fence. The floor is 4 \times 9 = 36 square meters, and 36 \div 9 = 4; the fence is 9 + 4 + 9 + 4 = 26 meters.
- Length plus width is 8: 7 by 1 with area 7, 6 by 2 with area 12, 5 by 3 with area 15, and 4 by 4 with area 16 square meters.
- 18 by 1 with perimeter 38 feet, 9 by 2 with perimeter 22 feet, and 6 by 3 with perimeter 18 feet. Widths 4 and 5 do not share out 18 evenly, and a width of 6 gives the 6 by 3 rectangle again.
- 24 by 1 with perimeter 50 feet, 12 by 2 with 28 feet, 8 by 3 with 22 feet, and 6 by 4 with 20 feet. A width of 5 does not share out 24 evenly.
- Both hold 32 square meters: 4 \times 8 = 32, and 2 \times 16 = 2 \times 10 + 2 \times 6 = 20 + 12 = 32. Pen A needs 8 + 4 + 8 + 4 = 24 meters of fence and pen B needs 16 + 2 + 16 + 2 = 36, so pen B needs 12 meters more.
- Both need 24 meters: 10 + 2 + 10 + 2 = 24 and 7 + 5 + 7 + 5 = 24. The areas are 2 \times 10 = 20 and 5 \times 7 = 35 square meters, so the 7 by 5 pen gives 15 square meters more.
- 16 by 1 with perimeter 34 centimeters, 8 by 2 with perimeter 20 centimeters, and 4 by 4 with perimeter 16 centimeters. Widths 3 and 5 do not share out 16 evenly.
- Length plus width is 10: 9 by 1 (area 9), 8 by 2 (16), 7 by 3 (21), 6 by 4 (24) and 5 by 5 (25). The most room is the 5 by 5 square, 25 square meters.
- The 6 by 5 pen, with 30 square meters. Length plus width is 11: 10 by 1 gives 10, 9 by 2 gives 18, 8 by 3 gives 24, 7 by 4 gives 28, and 6 by 5 gives 30.
- 10 feet by 8 feet, with 80 square feet. Length plus width is 9 panels. In feet the choices are 16 by 2 (32), 14 by 4 (56), 12 by 6 (72) and 10 by 8 (80).
- The 6 by 5 rectangle, with 22 feet of tape. The choices are 30 by 1 (62 feet), 15 by 2 (34 feet), 10 by 3 (26 feet) and 6 by 5 (22 feet); a width of 4 does not share out 30 evenly.
- The 7 by 7 square, with 49 square meters. Length plus width is 14, and the areas going down the list are 13, 24, 33, 40, 45, 48 and 49 for 13 by 1 up to 7 by 7.
- The 40 by 1 strip, with 40 + 1 + 40 + 1 = 82 feet of edge. The other rectangles are 20 by 2 (44 feet), 10 by 4 (28 feet) and 8 by 5 (26 feet).
- (a) 16 panels: the edge is 10 + 6 + 10 + 6 = 32 feet, and 32 \div 2 = 16. (b) 60 foam squares: 6 \times 10 = 60. (c) 6 cans: 60 \div 10 = 6.
- No. A 7 by 1 rectangle has perimeter 16 and area 7; a 3 by 3 square has the shorter perimeter 12 but the larger area 9.
- One answer is 4 meters by 3 meters. The 9 by 1 pen needs 20 meters of fence and holds 9 square meters; the 4 by 3 pen needs 14 meters and holds 12 square meters.
- A width of 0 means the two long sides lie on top of each other. The fence still adds to 6 + 0 + 6 + 0 = 12, but nothing is inside: the area is 6 \times 0 = 0, so it is not a pen at all.
- Length plus width is 26 \div 2 = 13. A square needs two equal whole numbers adding to 13, and 6 + 6 = 12 while 7 + 7 = 14, so there is none. The pens are 12 by 1 (12 square meters), 11 by 2 (22), 10 by 3 (30), 9 by 4 (36), 8 by 5 (40) and 7 by 6 (42), so 7 by 6 gives the most room, 42 square meters, with sides that differ by 1.
- The area counts squares, and the same 18 squares are used, so it stays 18 square feet. The perimeter counts the square edges on the outline. The 18 squares have 4 \times 18 = 72 edges, and every place two squares touch hides 2 of them. The strip has 17 touching places, hiding 34 edges and leaving 72 - 34 = 38, a perimeter of 38 feet. In the 6 by 3 rectangle each of the 3 rows has 5 touching places and each of the 6 columns has 2, so 15 + 12 = 27 places hide 54 edges and leave 72 - 54 = 18, a perimeter of 18 feet. Check: 6 + 3 + 6 + 3 = 18. Packing the squares closer hides more edges.