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What does it mean to find the Area (e.g. area of a circle)?by jaja1990
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#1
Feb612, 10:07 AM

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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.



#2
Feb612, 10:34 AM

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hi jaja1990!
area is a measure a measure gives a value µ(A) to any subset A, and obeys µ(A U B) = µ(A) + µ(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 µ(any rectangle) to be the product of the sides of that rectangle 


#3
Feb612, 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 todo list now. Can you explain a bit on: "a measure gives a value µ(A) to any subset A, and obeys µ(A U B) = µ(A) + µ(B)"? I understand what subset and union mean, but you didn't say what B is. Also, can you tell me how "obeys µ(A U B) = µ(A) + µ(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. 


#4
Feb612, 12:30 PM

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What does it mean to find the Area (e.g. area of a circle)?
hi jaja1990!
(i've now edited my previous post to correct that) we define its area to be the product of the sides … then we use µ(A U B) = µ(A) + µ(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 


#5
Feb812, 09:06 AM

P: 26




#7
Feb812, 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 µ(A U B) = µ(A) + µ(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. 


#8
Feb812, 11:59 AM

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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 


#9
Feb912, 09:52 AM

P: 26

I understand now, thank you for bearing with me!



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