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

[jaja1990]

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.

 

[tiny-tim]

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

 

[jaja1990]

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.

 

[tiny-tim]

hi jaja1990!

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.

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 µ(A U B) = µ(A) + µ(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 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

 

[jaja1990]

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 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) …

Can you tell me how this applies to a circle, for example?

 

[tiny-tim]

hi jaja1990!

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 …
http://protsyk.com/cms/wp-content/uploads/2011/12/QuadCircle_1.png

 

[jaja1990]

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.

 

[tiny-tim]

"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.

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]

I understand now, thank you for bearing with me!