Measuring area.

[stigofthedump]

Here's something totally unconnected to the coronavirus. Yay! A garden measures 20m x 40m so it's perimeter measures 120m (20+20+40+40m). It's area is 800 sq m (20 x 40m). The neighbour's garden measures 30m x 30m so it's perimetre also measures 120m. It's area is 900 sq m (30 x 30). Why is the area of the 2nd garden greater than the area of the 1st garden when the distance around each garden is the same? I have been trying to find an explanation for this for years. Can anyone explain it?

 

[MarkFL]

Suppose a rectangle has a fixed perimeter \(P\). Then if \(b\) is the base of the rectangle and \(h\) is the height, the area \(A\) is:

[MATH]A=bh[/MATH]
Now, we know:

[MATH]2b+2h=P\implies h=\frac{P}{2}-b[/MATH]
And so the area may be written:

[MATH]A=b\left(\frac{P}{2}-b\right)[/MATH]
The graph of this function is is a downward opening parabola, where the axis of symmetry, that is where the maximum value lies, is midway between the two roots, which is:

[MATH]b=\frac{0+\dfrac{P}{2}}{2}=\frac{P}{4}[/MATH]
This implies:

[MATH]b=h=\frac{P}{4}[/MATH]
So, what we've found is that for a rectangle having a given perimeter, the rectangle with the most area is a square.

 

[lev888]

Calculate the area of a circular garden with the same perimeter

 

[Steven G]

The area of a rectangle is b*h and the perimeter is 2(b+h)

I can find other numbers that multiply out to b*h, like (2b) and (h/2). The perimeter of that rectangle is 2(2b+h/2) which is different from 2(b+h)

In fact for ANY k>0 (does NOT have to be a whole number. It can be 7/3, 1/5, 7sqrt(2), pi, etc), (kb)*(h/k) = b*h, yet the perimeter is 2(kb + h/k). That is there are an infinite number of perimeters whose area is b*h.

A specific area of a rectangle does not correspond to a unique perimeter. Same for a given perimeter, the area could vary. Suppose you have 100ft of fencing and you want to build a rectangular dog pen. You can build it 1ft by 49ft using all 100ft of fencing and get an area of 49 sqft, or you can build it 20ft by 30ft using all 100 ft of fencing and get an area of 600sqft. 60 sqft is much much larger than 49 sqft and it used the same amount of fencing. You need to think about this deeply and settle it in your mind.

 

[Steven G]

Watch this video here!

 

[stigofthedump]

Suppose a rectangle has a fixed perimeter \(P\). Then if \(b\) is the base of the rectangle and \(h\) is the height, the area \(A\) is:

[MATH]A=bh[/MATH]
Now, we know:

[MATH]2b+2h=P\implies h=\frac{P}{2}-b[/MATH]
And so the area may be written:

[MATH]A=b\left(\frac{P}{2}-b\right)[/MATH]
The graph of this function is is a downward opening parabola, where the axis of symmetry, that is where the maximum value lies, is midway between the two roots, which is:

[MATH]b=\frac{0+\dfrac{P}{2}}{2}=\frac{P}{4}[/MATH]
This implies:

[MATH]b=h=\frac{P}{4}[/MATH]
So, what we've found is that for a rectangle having a given perimeter, the rectangle with the most area is a square.

WOW! Sounds impressive but I struggled to pass O level Maths so......it's "hand-over-head" for me. Thanks anyway.

 

[stigofthedump]

The area of a rectangle is b*h and the perimeter is 2(b+h)

I can find other numbers that multiply out to b*h, like (2b) and (h/2). The perimeter of that rectangle is 2(2b+h/2) which is different from 2(b+h)

In fact for ANY k>0 (does NOT have to be a whole number. It can be 7/3, 1/5, 7sqrt(2), pi, etc), (kb)*(h/k) = b*h, yet the perimeter is 2(kb + h/k). That is there are an infinite number of perimeters whose area is b*h.

A specific area of a rectangle does not correspond to a unique perimeter. Same for a given perimeter, the area could vary. Suppose you have 100ft of fencing and you want to build a rectangular dog pen. You can build it 1ft by 49ft using all 100ft of fencing and get an area of 49 sqft, or you can build it 20ft by 30ft using all 100 ft of fencing and get an area of 600sqft. 60 sqft is much much larger than 49 sqft and it used the same amount of fencing. You need to think about this deeply and settle it in your mind.

Hmmm. Need to think about that one. and watch the video. thanks.

 

[tkhunny]

WOW! Sounds impressive but I struggled to pass O level Maths so......it's "hand-over-head" for me. Thanks anyway.

This makes no sense. It is NOT beyond your capacity. Stop telling yourself that it is. Just look at one step at a time. No need to draw in the whole picture and hope you absorb it all.

The fixed perimeter is P. Any time one mentions "P", we are talking about the total perimeter. What is 2*P? That's two trips around the perimeter of the garden. We also get to know that P = 120 m. Great. What's ½*P? That's half way around the garden. It also happens to be ½ * 120 m = 60 m. It's just slight abstraction. If we don't know that P = 120 m, we can still talk about "P" as if we did know its value.

 

[Dr.Peterson]

Here's something totally unconnected to the coronavirus. Yay! A garden measures 20m x 40m so it's perimeter measures 120m (20+20+40+40m). It's area is 800 sq m (20 x 40m). The neighbour's garden measures 30m x 30m so it's perimetre also measures 120m. It's area is 900 sq m (30 x 30). Why is the area of the 2nd garden greater than the area of the 1st garden when the distance around each garden is the same? I have been trying to find an explanation for this for years. Can anyone explain it?

You're looking for the answer to a "why" question. That can be hard to answer, because "why" means many things. Ultimately, it requires a dialogue, not just a single answer.

But one answer is, just imagine making a loop of string of a particular length (a perimeter). You can lay that string out in many different ways -- a circle, a square, a snaky thing -- and if you stop to think about, it's obvious that they can't all have the same area. At the extreme, just pull it into a long double line looping around two nails, say, and it will contain an area of zero.

So the real question is, why would you think that figures with the same perimeter should have the same area at all?

Similarly, in reverse, think about making shapes of the same area, by rearranging a set of paper squares in various ways. Some will have a larger perimeter than others, though the area is the same -- for example 9 squares in a row make a 1 by 9 rectangle with perimeter 20; but in a 3 by 3 square, the perimeter is only 12. Why? When you put two squares next to one another, the length of the shared edges is no longer part of the perimeter. So the more "inside" there is (that is, the more edges are used up that way), the less "outside" there is (smaller perimeter). A shape that looks more like a circle will have a smaller perimeter for a given area (and a larger area for a given perimeter).

 

[JeffM]

Cut out of paper a bunch of squares one meter by one meter.

Let's go to your garden. i start covering the garden with square of paper. I put down along one edge. That takes 40 squares. Then I lay down 40 more adjacent to them. I do that 20 times. I add up the squares and find there are 800. I do the same thing with the neighbors garden. I count up the squares needed to cover that garden and find it to 900. So, in one respect, this is simply a matter of counting.

But what about multiplication.

[MATH]40 \times 20 = (30 + 10)(30 - 10) = 900 + 300 - 300 - 100 = 800.[/MATH]
That negative number reduces our result, which is exactly what we would expect.

 

[stigofthedump]

The area of a rectangle is b*h and the perimeter is 2(b+h)

I can find other numbers that multiply out to b*h, like (2b) and (h/2). The perimeter of that rectangle is 2(2b+h/2) which is different from 2(b+h)

In fact for ANY k>0 (does NOT have to be a whole number. It can be 7/3, 1/5, 7sqrt(2), pi, etc), (kb)*(h/k) = b*h, yet the perimeter is 2(kb + h/k). That is there are an infinite number of perimeters whose area is b*h.

A specific area of a rectangle does not correspond to a unique perimeter. Same for a given perimeter, the area could vary. Suppose you have 100ft of fencing and you want to build a rectangular dog pen. You can build it 1ft by 49ft using all 100ft of fencing and get an area of 49 sqft, or you can build it 20ft by 30ft using all 100 ft of fencing and get an area of 600sqft. 60 sqft is much much larger than 49 sqft and it used the same amount of fencing. You need to think about this deeply and settle it in your mind.

Thanks for your help. Am not sure what K means but I will take your advice and have a good think about your explanation. I should have known there wouldn't be a simple answer to my question

 

[stigofthedump]

The graph of this function is is a downward opening parabola, where the axis of symmetry, that is where the maximum value lies, is midway between the two roots, which is:

This is what I don't understand ............but i will spend more time studying your explanation. Thanks for helping.

 

[stigofthedump]

Cut out of paper a bunch of squares one meter by one meter.

Let's go to your garden. i start covering the garden with square of paper. I put down along one edge. That takes 40 squares. Then I lay down 40 more adjacent to them. I do that 20 times. I add up the squares and find there are 800. I do the same thing with the neighbors garden. I count up the squares needed to cover that garden and find it to 900. So, in one respect, this is simply a matter of counting.

But what about multiplication.

[MATH]40 \times 20 = (30 + 10)(30 - 10) = 900 + 300 - 300 - 100 = 800.[/MATH]
That negative number reduces our result, which is exactly what we would expect.

Thanks for your help however I can't understand what the 900, 300 & 100 represent.

 

[stigofthedump]

Here's something totally unconnected to the coronavirus. Yay! A garden measures 20m x 40m so it's perimeter measures 120m (20+20+40+40m). It's area is 800 sq m (20 x 40m). The neighbour's garden measures 30m x 30m so it's perimetre also measures 120m. It's area is 900 sq m (30 x 30). Why is the area of the 2nd garden greater than the area of the 1st garden when the distance around each garden is the same? I have been trying to find an explanation for this for years. Can anyone explain it?

The number of replies has been more than I expected with lots of explanations. I have more than enough to work through now so I feel my query has been answered. Thanks to everyone who replied.

 

[lev888]

Thanks for your help however I can't understand what the 900, 300 & 100 represent.

Those are the result of the multiplication (30+10)(30-10). Do you know how to do it? 30*30-30*10+...

 

[lev888]

Thanks for your help. Am not sure what K means but I will take your advice and have a good think about your explanation. I should have known there wouldn't be a simple answer to my question

k is a variable used to show that we can construct many rectangles with area bh.
If instead of b we use kb and instead of h we use h/k, the resulting rectangle will be different, but will still have area bh:
(kb)*(h/k) = b*h

 

[Steven G]

Thanks for your help. Am not sure what K means but I will take your advice and have a good think about your explanation. I should have known there wouldn't be a simple answer to my question

K is any positive number. 9*6=54. K=3: (9/3)*(6*3) = 3*18=54. K=2: (6/2)*(9*2) = 3 *18 = 54. K= 4: (9/4)*(6*4) = 2.25*24 = 54. This is all saying that 9ft*6ft = 3ft*18ft = 2.25ft*24ft = 54sqft.