Why is there more room in a square room than there is in a rectangular room when the perimeter is the same in both rooms?

Question

Why is there an extra square foot in a square room with dimensions of $13×13$ and one less square foot in a room with dimensions of $14×12$? The perimeter for both rooms is the same (52 foot). I'm not asking to see formulas giving me the same results I already know - I'd prefer a more intuitive explanation.

Answer 1

Let $\;R\;$ be a rectange with sides $\;x,y\;$ s.t $\;x+y=k\;,\;\;k\in\Bbb N\;$ fixed, let us find the maximum of $\;f(x,y):=xy$:

$$x+y=k\implies y=k-x\implies g(x)=f(x,k-x)=x(k-x)$$

From here:

$$g'(x)=k-2x=0\iff x=\frac k2\implies y=\frac k2$$

and the rectangle is in fact a square. you now prove that in the above we really get a maximum point.

Another more basic way with the same notation:

$$g(x)=-x^2+kx , \;\;\text{and this is a parabola that open downwards}$$

and thus its vertex is a maximal point. But the vertex is given by

$$\left(\;-\frac k{-2}\;,\;\;-\frac\Delta{-4}\;\right)=\left(\;\frac k2\;,\;\;\frac{k^2}4\;\right)$$

and again $\;x=\frac k2\;$ and etc.

Answer 2

If I may rephrase the problem, what is the maximum area $A$ of a rectangle of given perimeter $P$ ?

Let us call $a$ and $b$ the sizes of the rectangle. We then have $$A=a \times b$$ $$P=2\times(a+b)$$ Extract $b$ from $P$; this gives $b=\frac P2 -a$; then $$A=a \times\Big(\frac P2 -a\Big)$$ Now, compute the derivative with respect to $a$; so $$\frac{dA}{da}=\frac{P}{2}-2 a$$ So, an extremum is obtained for $a=\frac P4=b$ (this means a square). The second derivative test confirms that this is a maximum.

Answer 3

From the comments:

I can take that rectangle, cut the length off, add it to the width, and get the same area as the square.

Well, not quite.

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While I wait to properly understand the question, here are some more "ridiculous algebraic formulas". Suppose the size of the square is $a$. Then the dimensions of the rectangle must be $a+b$ and $a-b$ for some $b$. Its area is $(a+b)\times(a-b)=a^2-b^2$, which is less than the area of the square, $a^2$.

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From the later comments:

Mathematics is a man made concept. Everything has a reason for being the way it is. . . There was a reason the inventors of mathematics created it the way they did. . . A human created the formula to determine the area and the perimeter, etc of something.

I see! You're asking the question "Why does the square have larger area than the rectangle?" in the sense of something like "Why is the speed limit only 20 miles per hour in this podunk town?" Someone was responsible for that speed limit. They could have chosen otherwise, but they chose this. Why?

But here I don't think people had that much choice in the matter. Area is just a name for the number of little squares that can fit inside a given shape. Perimeter is a name for the number of little line segments that can go along the boundary of the shape. Those would still be there even if we hadn't given those concepts a name, just as a lemon would have a colour and a taste even if we didn't have words for yellow and sour. I mean, you can draw a $4\times2$ rectangle and a $3\times3$ square on a piece of graph paper and count the number of little squares and the number of little lengths on the boundary by hand without using any formulas. Go ahead, try it right now! What part of that feels invented, or man-made, or arbitrary? Did someone else choose those numbers, when you counted them yourself?

It's kind of implicit in your original framing of the question, too, isn't it? You've built a $14\times12$ foot room and a $13\times13$ foot room, and you find that the $13\times13$ room has more space in it. And you want to know why someone defined the formulas so that it works out this way. The idea that some mathematicians in antiquity are responsible, that if only they had defined area and perimeter differently it would somehow change the amount of stuff you can fit into your rooms, well, it doesn't make any sense to me.

Answer 4

Constant perimeter does not imply constant area, because the later depends on $A+B$ while the former depends on $AB$. And recaling the AM-GM inequality:

$$ \sqrt{A B } \le \frac{A+B}{2} \implies S=A B \le \frac{(A+B)^2}{4}=\frac{P^2}{16}$$

The equality happens when (and only when) $A=B$

Answer 5

I'll try to give an intuitive answer. For the rigorous mathematical argument you have Claude's answer.

Let's imagine we build our rectangles and squares from small $1\times 1$ blocks.

To build a rectangle with area $4$, we need to join $4$ small squares consecutively. You'll have two internal squares, in which two sides are belong to the perimeter, and two external squares, in which three sides belong to the perimeter.

So you'll have a perimeter of: $2 \times 2 + 2\times 3 = 10$

Now, to build a square with area $4$, you put $2$ small squares on top of another two small squares. All of these will have $2$ sides that contribute to the perimeter.

The perimeter will be: $4 \times 2 = 8$.

This same argument can be increased to any size square/rectangle. Any rectangle will always have more of these blocks exposed to the outside than a square of the same area.

This proves that a rectangle will always have a larger perimeter than a square with the same area. This implies that if a rectangle and a square have the same perimeter, the rectangle must have a smaller area.

Answer 6

This is both in response to the comments made in Rahul's answer, and also as an additional graphical proof.

In addition, let me emphasize that to mathematicians, "Why" something is true is explained in mathematical proof. Explaining things via formula and pictures and such is not a "How" but is indeed a "Why". Also let me emphasize that when beginning anything it is important to remember the definitions. Area of a shape is loosely defined as "how many unit squares can fit in a shape", and for rectangles in particular it is explicitly defined as $A\times B$ where $A$ and $B$ are the lengths of the sides.

Claim: All rectangles of a given perimeter have less than or equal area to that of the square of the same perimeter. Equivalently, a square of a given perimeter is of greater area than all rectangles of equal perimeter where the rectangle is not a square.

Without loss of generality let the rectangle we are comparing to be longer than it is tall. Having started with a square of side length $B$, choose some positive value of $x\leq B$ and relabel the side length as $B=A+x$ where $A = B-x$

Make a horizontal slice a distance of $x$ away from the top extending across the entire square and a vertical slice a distance of $x$ away from the right which extends from the top to the previous slice. By removing the upper-right piece you effectively reduce the area. You may then move the upper-left piece and rotate it to fit nicely with the lower piece. The resulting formed rectangle is of the same perimeter as the original square, however the area is smaller.

As such the area of the blue square is $A^2 + 2Ax + x^2$ which is strictly greater than the area of the red square which is $A^2 + 2Ax$ while the perimeter remain the same.

By changing the value of $x$, you are able to form any rectangle of given perimeter, and if $x=B$ then you get a line segment which has zero area. $\square$

If you are looking for a simpler answer, the "why" is because area and perimeter measure two completely different things. If you are looking for a more philosophical answer, then I would direct you to read Plato and Pythagoras.

Answer 7

The perimeter and the area of a shape have little or no relation to one another.

Cut yourself a length of string and tie it together to make a loop. You can lay this out on the ground so that it surrounds many different sized areas.

There will be a maximum possible area. This is a famous problem known as the Isoperimetric Problem. It turns out that the maximum enclosed area is given by making a circle.

But what about the smallest possible enclosed area? Well, if you use "mathematical string", i.e. a one dimensional object with no thickness and with perfect "bendability", then you can \contrive a shape that encloses an area as small as you like. Think about a long, thin rectangle. If you want the enclosed area to be less then make the rectangle longer and thinner.

Mathematically: Let the length of string be $p$ (for perimeter), and make a rectangle with width $w$ and thickness $t$. The perimeter is, by definition, given by $p = 2w+2t$. The area will, by definition, be given by $a=w t$. The perimeter is a constant and so we can rearrange $p=2w+2t$ to give $$w=\tfrac{1}{2}p -t $$ It follows that the area of the rectangle is given by $$\begin{eqnarray*} a &=& wt \\ \\ &=& \left(\tfrac{1}{2}p -t\right)t \\ \\ &=& \tfrac{1}{2}pt - t^2 \end{eqnarray*}$$ The area $a=\tfrac{1}{2}pt - t^2$ has no minimum. No matter what the perimeter $p$ is, as the thickness $t \to 0$, the area $a \to 0$. As the thickness $t \to 0$, the width $w \to \tfrac{1}{2}p$.

What about the maximum possible area for a rectangle? (A circle encloses the maximum possible area.) Completing the square on $a=\tfrac{1}{2}pt - t^2$ gives $$a = \tfrac{1}{16}p^2-\left(t-\tfrac{1}{4}p\right)^{\! 2}$$

This shows that, in the case of the rectangle, the maximum enclosed area is given by making by $t-\tfrac{1}{4}p=0$, i.e. by making the thickness one quarter of the perimeter. This means we have made a square. In such a case, the maximum area is $\tfrac{1}{16}p^2$.

TLDR: There is no general connection between the perimeter and the enclosed area. You can make an enclosed area as small as you like for any given perimeter. There is, however, a maximum possible area for any given perimeter; this is formed by a circle.

Answer 8

My attempt at making it as simple as possible:

Suppose you have a rectangle that isn't a square. Then it's longer in one dimension than the other.

Now take a little bit (say 1 inch) off of the long sides and add it to the short sides.

Since you added 1 inch to two sides, and removed 1 inch from two sides, the perimeter stays the same.

By adding 1 inch to the short sides, you added (1 inch) * (long side) to the area. By subtracting 1 inch from the long sides, you subtracted (1 inch) * (short side) from the area. Since the long sides were longer than the short sides to begin with, you added more area than you subtracted.

Therefore you can always increase a rectangle's area, without changing its perimeter, by transferring length from the long sides to the short sides (as long as the amount of length that you transfer isn't more than half the difference between them).

The only time this isn't possible is when all of the sides are the same length — that is, when the rectangle is a square. Therefore, a square maximizes area for a given perimeter.

Answer 9

Take a 13x13 square, and for convenience's sake, draw it on a 1x1 grid, so it takes up 13x13 squares.

Pick one corner of the square (let's say top left), and remove the 1x1 square in that corner from the big square. The shape you have now (big-square-minus-little-square) has the same perimeter as the big square that you started with, but obviously its area is 1 square less than the area of the big square that you started with.

Now remove another square, adjacent to the one that you first removed (let's say top almost-left). You now have a 13x13 square from which you removed a 2x1 rectangle. The perimeter is still the same, but the area has been diminished by two.

Keep removing 1x1 squares along the same edge (top edge), until there is only one 1x1 square left on that edge. You now have a 13x13 square from which you removed a 12x1 rectangle, without changing its perimeter. You can also think of it as a 13x12 rectangle with a single 1x1 square hanging off one corner (at the top of the top right corner).

Now take that one extra 1x1 square and move it from the edge it's on (top edge) to its adjacent edge (right edge), so that now you have a 13x12 rectangle with a single 1x1 square hanging off the same corner, but in the other direction (at the right of the top right corner). You can also think of it as a 14x12 rectangle with a 11x1 rectangle missing. In this step, you once again did not change the perimeter, and you also did not change the area.

Now, starting from the 1x1 square that's hanging off the corner, add one missing 1x1 square back next to it (just below it). You now have a 14x12 rectangle with a 10x1 square missing. In this step, you did not change the perimeter, but you increased the area by 1.

Keep adding 1x1 squares back until you have a full 14x12 rectangle. Every time you add a 1x1 square back, you are increasing the area by 1, without changing perimeter.

Now look at what you did over all: you removed 13-1=12 squares off the top, but you added 12-1=11 squares on the right. Therefore, the area decreased overall from start to finish.

If you think about how this process works, you will see that if you start with a square, you will always end up removing more 1x1 squares from the top than you add on the right, and therefore the area will always decrease overall. (Try it for a few different size squares to see why.)

This process even works if you start with a rectangle, as long as you always remove squares from a long edge and put them back on a short edge. For example, you can start with a 13x13 square and go to 14x12 (removing some area) and then to 15x11 (removing some more area). This lets you start with a square and change the edges by more than 1, and still be sure that some area will be lost.

Answer 10

Let's says the rectangle has sides of length $A$ and $B$ and the square has sides of length $C$. Because they have the same perimeter we can say $$ A + B = 2C \, . $$ Now let's assume that in the rectangle, $A$ is the bigger side and $B$ is the smaller side. Which means $A$ will be bigger than $C$ and $B$ will be smaller than $C$. That means for some non-negative number $x$
\begin{align*} A = C + x\\ B = C - x \end{align*} so the area of the rectangle will be $$ AB = (C+x)(C-x) = C^2 - x^2 $$ so the area of the square will be greater than that of the rectangle by the square of difference of the sides length.

Answer 11

Robert, to keep it simple think of perimeter as 'Exposed Surface'. Perhaps a visual illustration of 'why' will be of more use to you than an algebraic one?

Start with this:

Then with that in mind, this:

Notice how 'Area' and 'Total Surface' stay constant throughout the shape change (folding) process. Modifying the shape of an object directly influences the ratio of 'Exposed' & 'Concealed' surfaces (measured in terms of discrete units). Therfore the 'perimiter' of an object can be modified significantly without the need to change its area.

Also notice how the more compact we make the object (by reducing the amount of exposed surface, and therfore increasing the concealed surface) the more square like it becomes, this is because the square is the most compact form of rectangle, it has the least amount of exposed surface possible for a 'rectangular shaped object'.

As for the units used, they don't really matter, we could be referring to (m), (cm), (mm), etc. down to atomic distances if you like:

Either way the concept is the same; you break the object down to discrete units, and from there you can analyse the relationship between shape, area, and perimeter.

Answer 12

The reason the area of a rectangle and the area of a square are not equivalent (assuming the perimeter is the same) is due to the fact that the perimeter and area are inextricably linked. A perimeter defines an area and an area only exists when a perimeter is defined.

It turns out that the configuration of the perimeter directly impacts the value of the area. If you have 12 feet of fencing you can make a square with 3 feet of fencing on each side resulting in 9 square feet of area (3x3 = 9). You can also configure that fencing in a rectangular shape with a width of 5 and a height of 1. This still has a perimeter of 12 feet, but only an area of 5 square feet (5x1 = 5).

If you infinitely shrink the height of the rectangle to zero you will end up with the top and the bottom of the rectangle being 6 feet in length with zero height and an area of 0 square feet (6x0 = 0). This is because the top and the bottom are touching and there is no area enclosed.

This applies to all possible configurations of the perimeter and the area of a rectangle and a square lie along a spectrum of what you might call the efficient closure of an area. For any given perimeter it turns out that a circle encloses the greatest area. This is because every section of the perimeter is laid out in the most 'efficient' manner.

We can see this in nature in the physical world if you're willing to jump into 3 dimensions for a moment. In 3 dimensional space area is akin to volume and perimeter is akin to surface area. When you look at a star, or a water droplet where every atom is under the influence of gravitational attraction or hydrogen bonding, they try to get as close together as physically possible. This results in a spherical shape so that there is no 'inefficiently' used space.

I had the same question a long time ago, and this line of thinking is what made the most sense to me.