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What does it mean to find the Area (e.g. area of a circle)?

by jaja1990
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jaja1990
#1
Feb6-12, 10:07 AM
P: 26
What does it mean to find the area? I've read somewhere and the person says, it means to find the space enclosed, but I still don't know what that means. I understand what area intuitively means, but not logically.
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tiny-tim
#2
Feb6-12, 10:34 AM
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hi jaja1990!

area is a measure

a measure gives a value (A) to any subset A, and obeys (A U B) = (A) + (B), for any two subsets A and B which do not overlap

(see http://en.wikipedia.org/wiki/Measure_(mathematics) for more details)

it could be area, or probability, or cost, or

for area, we define (any rectangle) to be the product of the sides of that rectangle
jaja1990
#3
Feb6-12, 12:10 PM
P: 26
Thank you, that was a lovely answer.

I won't be able to fully understand the topic in the link yet, but it's on my to-do list now.

Can you explain a bit on: "a measure gives a value (A) to any subset A, and obeys (A U B) = (A) + (B)"?
I understand what subset and union mean, but you didn't say what B is. Also, can you tell me how "obeys (A U B) = (A) + (B)" applies to finding the area of a rectangle?

I hope I'm not being boring by asking these questions and reading more myself. Right now, because I'm short on time, I'm just trying to get a general idea, not delve deeply and look for exact answers.

tiny-tim
#4
Feb6-12, 12:30 PM
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What does it mean to find the Area (e.g. area of a circle)?

hi jaja1990!
Quote Quote by jaja1990 View Post
Can you explain a bit on: "a measure gives a value (A) to any subset A, and obeys (A U B) = (A) + (B)"? I understand what subset and union mean, but you didn't say what B is.
ooh, i should have said that B also had to be a subset, with no overlap (A intersection B is empty)

(i've now edited my previous post to correct that)
Also, can you tell me how "obeys (A U B) = (A) + (B)" applies to finding the area of a rectangle?
finding the area of a rectangle isn't a problem

we define its area to be the product of the sides

then we use (A U B) = (A) + (B) to define the area of any other shape (in the same way that the ancient greeks did)

we fill out the shape with rectangles, and add up the areas of the rectangles
jaja1990
#5
Feb8-12, 09:06 AM
P: 26
finding the area of a rectangle isn't a problem

we define its area to be the product of the sides

...

we fill out the shape with rectangles, and add up the areas of the rectangles
Why isn't it a problem for a rectangle, while it is for others? Ummm... is it because we just take the area of a rectangle to find other areas?

then we use (A U B) = (A) + (B) to define the area of any other shape (in the same way that the ancient greeks did)
Can you tell me how this applies to a circle, for example?
tiny-tim
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Feb8-12, 09:16 AM
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hi jaja1990!
Quote Quote by jaja1990 View Post
Why isn't it a problem for a rectangle, while it is for others? Ummm... is it because we just take the area of a rectangle to find other areas?
yup!
Can you tell me how this applies to a circle, for example?
like this
jaja1990
#7
Feb8-12, 09:25 AM
P: 26
I understand how "we fill out the shape with rectangles, and add up the areas of the rectangles" applies to a circle, I was asking about:-

"then we use (A U B) = (A) + (B) to define the area of any other shape (in the same way that the ancient greeks did) …"

Specifically, I don't understand how we choose "A" and "B", I don't know how their values would look like for a circle.
tiny-tim
#8
Feb8-12, 11:59 AM
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Quote Quote by jaja1990 View Post
"then we use (A U B) = (A) + (B) to define the area of any other shape (in the same way that the ancient greeks did) "

Specifically, I don't understand how we choose "A" and "B", I don't know how their values would look like for a circle.
well, we need a lot more letters than that!

A B C D are the areas of the 1st 2nd 3rd 4th rectangles

we add up the areas of as many rectangles as are needed, to get whatever degree of accuracy we want
jaja1990
#9
Feb9-12, 09:52 AM
P: 26
I understand now, thank you for bearing with me!


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