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

  1. 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. jcsd
  3. tiny-tim

    tiny-tim 26,054
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    hi jaja1990! :smile:

    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 :wink:
     
    Last edited: Feb 6, 2012
  4. 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 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.
     
  5. tiny-tim

    tiny-tim 26,054
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    hi jaja1990! :smile:
    ooh, i should have said that B also had to be a subset, with no overlap (A intersection B is empty) :redface:

    (i've now edited my previous post to correct that)
    finding the area of a rectangle isn't a problem …

    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
     
  6. 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?

    Can you tell me how this applies to a circle, for example?
     
  7. tiny-tim

    tiny-tim 26,054
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    hi jaja1990! :smile:
    yup! :biggrin:
    like this :wink:
    [​IMG]
     
  8. 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.
     
  9. tiny-tim

    tiny-tim 26,054
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    well, we need a lot more letters than that! :biggrin:

    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 :smile:
     
  10. I understand now, thank you for bearing with me! :biggrin:
     
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