I really am no mathematician but let me give it a go.
For any given perimeter, a circle will always be the most efficient way to enclose the biggest possible area. Let's pretend you have a piece of string, the floor is covered with Maltesers, and I'll give you the Maltesers as long as they're enclosed by your string. Assuming you like chocolate, you'd want to arrange the piece of string as much like a circle as possible to cover as many Maltesers as possible. You wouldn't want to arrange your string "like a worm" (i.e. as a long thin shape)—you'd probably only get single Maltesers that fit within the worm. Imagine if you were able to make your string infinitely like a worm (i.e. maximally long and thin for that given perimeter): its area would be near zero and you wouldn't get any Maltesers at all.
The same holds true for rectangles. The closer a rectangle is to a square, the more efficient it will be at enclosing a big area—in other words, the bigger its area is. Imagine the same piece of string, but instead of making circles or long thin shapes, it's making rectangles. Given the same perimeter, the longer and thinner the rectangle is, the smaller its area will be. And that's what the chart is showing you.
My explanation is to think of your perimeter all straightened into a single line, but made up of meccano type pieces so you can click it into all possible rectangles. Long and thin or short and fat. You might well notice that you only need half of your long perimeter to make two sides of the rectangle. You can call the longer one L and the shorter one W length and width. Or call them whatever you like. Your perimeter will always be 2 x L + 2 x W or, more elegantly, 2(L + W). Your area will always be L x W, or more elegantly LW. To make the area as big as possible you need L and W to be as close to each other in value as you can make them. Conversely, to make the area as small as possible make W (or L) as small as you can, say 1.
For example a rectangle of perimeter 20 could have its smallest area with length 9 and width 1 (area 9 squares) or its largest with length 5 and width 5 (area 25 squares).
A practical application of this comes up in architecture. If you’re designing an office you generally want narrow floor plates (long rectangles instead of squares). Usually you want to minimize east and west glass and maximize south and north.
If you imagine you are told you can have 900square feet per floor and the most important criteria is that every desk have a view, natural light from multiple directions and cross ventilation. So ( extreme example but for ease of numbers ) you make your building 90’ long by 10’ wide. Every desk faces the south windows and circulation is a long the north. Every desk is within a few feet of both north and south windows. That building has 200’ of exterior wall (perimeter).
On another project the most important criteria are cost and energy efficiency. And same requirement for 900 square feet per floor. Energy is lost through the exterior walls , especially the glassy walls of an office building. Exterior wall is also costly to build. So in this case you’re going to make your plan 30’x30’. (I’m using small numbers for ease of math but imagine a huge square downtown office building and how a desk in the middle of that square will be so far from a window). The perimeter for this building is just 120’.
In real life you want to find some happy medium between these two. There are stairs, elevators bathrooms that can go into the middle of a more squareish bldg. And while less perimeter wall is less area to lose energy through in the second example you have 30’ of west glass rather than the 10’ in the first. It’s harder to control the heat coming through west glass so even though you reduced your overall perimeter you increased a part of the perimeter (west side) that has more energy consequences.
This is very helpful. I have to say I still find it rather counterintuitive that the same perimeter can enclose completely different areas, but thanks to Meccano and Maltesers I can at least now visualise it very nicely! Funnily enough I have just noticed that this question is covered in my daughter’s maths workbook, which is aimed at 8-9 year olds… which makes me think this should maybe have come up somewhere in my own education some time ago?! Maybe I was’t paying attention in that maths lesson!
Organizing a sailing class is truly next level Catherine! (my youngest son would sign up instantly:) Also, at the mention of Amazon I remembered a very fun memory resource we used when the kids were younger https://www.memorize.academy/blog/how-to-memorize-the-ten-longest-rivers-in-the-world. It is a white board animation story method and it works splendidly for anything from the "longest rivers of the world" to British monarch, countries and capitals, and the periodic table. Kyle has many videos on his blog section that you can access for free:)
I really am no mathematician but let me give it a go.
For any given perimeter, a circle will always be the most efficient way to enclose the biggest possible area. Let's pretend you have a piece of string, the floor is covered with Maltesers, and I'll give you the Maltesers as long as they're enclosed by your string. Assuming you like chocolate, you'd want to arrange the piece of string as much like a circle as possible to cover as many Maltesers as possible. You wouldn't want to arrange your string "like a worm" (i.e. as a long thin shape)—you'd probably only get single Maltesers that fit within the worm. Imagine if you were able to make your string infinitely like a worm (i.e. maximally long and thin for that given perimeter): its area would be near zero and you wouldn't get any Maltesers at all.
The same holds true for rectangles. The closer a rectangle is to a square, the more efficient it will be at enclosing a big area—in other words, the bigger its area is. Imagine the same piece of string, but instead of making circles or long thin shapes, it's making rectangles. Given the same perimeter, the longer and thinner the rectangle is, the smaller its area will be. And that's what the chart is showing you.
That is a brilliant explanation, thank you. And Maltesers are a very motivating choice of example!
JCX has it nicely.
My explanation is to think of your perimeter all straightened into a single line, but made up of meccano type pieces so you can click it into all possible rectangles. Long and thin or short and fat. You might well notice that you only need half of your long perimeter to make two sides of the rectangle. You can call the longer one L and the shorter one W length and width. Or call them whatever you like. Your perimeter will always be 2 x L + 2 x W or, more elegantly, 2(L + W). Your area will always be L x W, or more elegantly LW. To make the area as big as possible you need L and W to be as close to each other in value as you can make them. Conversely, to make the area as small as possible make W (or L) as small as you can, say 1.
For example a rectangle of perimeter 20 could have its smallest area with length 9 and width 1 (area 9 squares) or its largest with length 5 and width 5 (area 25 squares).
Which of those would hold more Maltesers??
A practical application of this comes up in architecture. If you’re designing an office you generally want narrow floor plates (long rectangles instead of squares). Usually you want to minimize east and west glass and maximize south and north.
If you imagine you are told you can have 900square feet per floor and the most important criteria is that every desk have a view, natural light from multiple directions and cross ventilation. So ( extreme example but for ease of numbers ) you make your building 90’ long by 10’ wide. Every desk faces the south windows and circulation is a long the north. Every desk is within a few feet of both north and south windows. That building has 200’ of exterior wall (perimeter).
On another project the most important criteria are cost and energy efficiency. And same requirement for 900 square feet per floor. Energy is lost through the exterior walls , especially the glassy walls of an office building. Exterior wall is also costly to build. So in this case you’re going to make your plan 30’x30’. (I’m using small numbers for ease of math but imagine a huge square downtown office building and how a desk in the middle of that square will be so far from a window). The perimeter for this building is just 120’.
In real life you want to find some happy medium between these two. There are stairs, elevators bathrooms that can go into the middle of a more squareish bldg. And while less perimeter wall is less area to lose energy through in the second example you have 30’ of west glass rather than the 10’ in the first. It’s harder to control the heat coming through west glass so even though you reduced your overall perimeter you increased a part of the perimeter (west side) that has more energy consequences.
This is fascinating, thank you so much for sharing Erin!
This is very helpful. I have to say I still find it rather counterintuitive that the same perimeter can enclose completely different areas, but thanks to Meccano and Maltesers I can at least now visualise it very nicely! Funnily enough I have just noticed that this question is covered in my daughter’s maths workbook, which is aimed at 8-9 year olds… which makes me think this should maybe have come up somewhere in my own education some time ago?! Maybe I was’t paying attention in that maths lesson!
Organizing a sailing class is truly next level Catherine! (my youngest son would sign up instantly:) Also, at the mention of Amazon I remembered a very fun memory resource we used when the kids were younger https://www.memorize.academy/blog/how-to-memorize-the-ten-longest-rivers-in-the-world. It is a white board animation story method and it works splendidly for anything from the "longest rivers of the world" to British monarch, countries and capitals, and the periodic table. Kyle has many videos on his blog section that you can access for free:)
Thank you Ruth, I did feel rather proud of myself! And that memory technique looks great, I will definitely be doing that with the children.
I am so glad you linked Emily Wilson on Substack! I love her translation but hadn’t found her here yet.
Very inspiring! We also follow this kind of timetable (no times, just things to get done) and it works so much better with younger kids.