Math question about area and perimeter?

Question

I was working with area and perimeter today, and I noticed that a rectangle can have an area of 8 sq. units and perimeter of 12 units. Then a square can have an area of 9 sq. units and perimeter of 12 units. What I don't understand is how can the perimeter be the same, but the area is different. If I were to cut a piece of string and tie it in a knot, wouldn't it fit around the same amount of blocks (representing area)? This is driving me crazy. I don't understand why it can be different. Can someone explain this to me so that I can sleep tonight?

I understand how to get perimeter and area, but I don't understand why if two objects have the same perimeter WHY they don't have to have the same area as well. If I had a string, and no matter what shape I put it in, it should enclose the same amount of space. So, area if the amount of space enclosed by the perimeter. If the perimeter is the same, why isn't the area?

I meant to say: I understand HOW to get perimeter and area, but I don't understand why if two objects have the same perimeter WHY they don't have to have the same area as well. If I had a string, and no matter what shape I put it in, it should enclose the same amount of space. So, area IS the amount of space enclosed by the perimeter. If the perimeter is the same, why isn't the area?

You're not understanding my question.
Item A has a length of 4 and width of 2.
Item A's perimeter is 12. Item A's area is 8
Item B has a length of 3 and width of 3.
Item B's perimeter is 12. Item B's area is 9

The perimeter of the two items is the same. Why isn't the area the same? How can the perimeter stay the same but the area it encloses be different sizes?

I thought area means the surface that the perimeter surrounds.

None of these are doing it for me. Forget it.

Fernandes. Thank you for adding light into my world. All I could think of was that the string was fully extended in the 1x4 rectangle as it would be also in the 3x3 square. But your illustration helped me to understand the difference. Bless you.

Answer 1

The area within the string can vary depending on the shape drawn.

Take the string which forms a 9sq unit square, Now take half of the string from point A to point B and then back from point B to Point A. This is almost a line with no space in between.

Now you'll notice that the perimeter of this shape is the same 12 units, but area is almost zero.

Answer 2

It's because perimeter is a sum and area is a product.

If you take (a + b) and get some number, and then multiply those (ab) you get a different number......obviously right?

Now take 2 other numbers (m + n), you CAN get the same result as (a + b) but not necessarily the same PRODUCT.

Perimeter is a sum, area is a product.

Here's an example:

5 + 2 = 8
5*2 = 10

6 + 2 = 8
6*2 = 12

Same sum...different product.

If you took a square, and then pulled the sides away from each other, and then imagine the top and bottom having to become narrower and narrower to stay 'attached' you aren't changing the area.....right? But you ARE changing the lengths of the sides.

Since you understand a little about the concepts, here's a little math that may help too.


Square:
Perimeter (p) = 2(l + w)
Area (a) = lw

Rect:
Perimeter (P) = 2(L + W)
Area (A) = LW

If p = P (same perimeters between the shapes)

--> 2(L + W) = 2(l + w)
--> L + W = l + w

This is deceiving because it looks like l = L and w = W, but NOT so.....as long as the SUM is the same it is valid:

10 + 2 = 12
6 + 6 = 12
7 + 5 = 12
4 + 8 = 12

etc etc.....do you see?

Now......area.....a = lw, and A = LW

You can see already that you don't need the same l, and L and w and W so it's easy to see that the products will be different.

Same numbers:

10*2 = 20
6*6 = 36
7*5 = 35
4*8 = 32

Very different results. You are performing different operations with different numbers....of course you will get different results, but the CONCEPT that you are working with shapes makes it deceiving in that it is a tangible object you can see, or feel.

That's why it's so hard for a lot of people to make that 'leap' to trust the math and not necessarily their *perception*.

Perception is deceiving. We have common sense built in to us to keep us safe from making mistakes because of that (the stove doesn't *appear* hot, but we just cooked dinner a little while ago so I *know* it's still hot), but it's a battle between those two. Making the leap to putting the trust in the math is a difficult one, but it's the key that unlocks where the beauty and creativity really is with math.

Does this help you at all? Hopefully so. Interesting question........very nice to think about!

Answer 3

What you describe with the piece of string is the perimeter!

Imagine a square, a real square like Trafalgar Square; if you walk around it you will walk its perimeter and never its area.

The area is the space inside a perimeter. In the case of Trafalgar square, we would have to walk from one end to the other on a straight line and come back, repeating this as many times as needed to cover the whole inside and that is why the area is measured in sq units, as you have to walk the straight line, let's call it a, for a certain n umber of times, let's call it be

so the area is a*b

and the perimeter is a+b+a+b = 2a+2b

In the case of a triangle we get b*h/2 for the area and a+b+c for the perimeter

a+b+c can be equal to 2a+2b without a*b being equal to b*h/2

this is true for any polygon

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Now I understand your issue and the answer is still the same.

Still imagine Trafalgar square and imagine it as a 3 by 4 units rectangle. The perimeter is 14 units and the area is 12 sq units

Now divide the rectangle in 2 triangles from one extreme point of the rectangle into the other. This new line measures 5 units (applying the Pythagoras theorem 3^2+4^2=5^2) and the area of the 2 triangles is half of the area of the rectangle, 6 sq units.

Now imagine the piece of string that only covers one of the triangles (a 3 by 4 by 5 string) and rearrange that piece of string to become a smaller Trafalgar Square (a 3 by 3 square).

As I previously said, walk this rearranged perimeter and tell me, what is the area you cover? Obviously it is not half of Trafalgar Square! It is more! Just because two 3 by 3 squares are bigger than one 3 by 4 square!

Answer 4

Why compare a rectangle and a square? Just take the case of a recangle having an area of 8 square units and it will be easy to understand the riddle. You can decrease the breadth of the rectangle to 1 and you have a length of 8 units, the area remains 8 sq.u. but what happens to the perimeter? It is now 18 units! Keep reducing the breadth till the breadth tends to zero, what will happen to the length and perimeter? It will tend to infinity!But area will remain the same within the mathematical accuracy limits or as per the uncertaity principle of Heisenberg .So numerically , since perimeter is addition of digits and area is multiplication , so their relationship will change accordingly and has to be established as per area and perimeter formula and not through common sense.A cetain object with same area but varying length and breadth will have different perimeters since it depends on length to breadth ratio of enclosed area.So, make it a part of your basic common sense and don't try to wrap strings around objects having same area but varrying length and breadth.Basic mathematics never suggests this! Hope you will have a good night's sleep now.

Answer 5

very simple - 2 x 4 gives perimeter of 12 with 8 sq ft area

3 x 3 gives perimeter of 12 with 9 sq ft area

it is the configuration

and 1 x 5 gives 12 perimeter and 5 sq feet


sleeeeeeeppppppppp weelllllllllllllllllllllllllllllllll!

Answer 6

How can U think dat 2 bodies having da same perimeterwill have da same area. Area means Len*Brdth. Perimetermeans the length of all da 4 sides. Suppose the length and breadth of a Sq. is x and the L and b of a Rect are a and b.
x^2=ab So, why should 4x=2a+2b.i.e. 2x=a+b
x^2=(a+b)^2/2-(a-b)^2/2
4x^2=(a+b)^2-(a-b)^2
(2x)^2= ''

Answer 7

the answer is simple
you are comparing two values of different dimensions which is obviously wrong
for example perimeter is single dimensional quantity(metre)
but area is two dimensional quantity(metre square)

Answer 8

please mention the length of rectangle