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

In summary, finding the area means to find the space enclosed by a shape. This can be done using a measure, which assigns a value to any subset and follows the rule of adding the values of two non-overlapping subsets to get the value of their union. This concept can be applied to different shapes, such as a rectangle or a circle, by dividing them into smaller rectangles and adding up their areas to get an approximation of the shape's area.
  • #1
jaja1990
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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.
 
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  • #2
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:
 
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  • #3
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.
 
  • #4
hi jaja1990! :smile:
jaja1990 said:
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) :redface:

(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
 
  • #5
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?
 
  • #6
hi jaja1990! :smile:
jaja1990 said:
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! :biggrin:
Can you tell me how this applies to a circle, for example?

like this :wink:
http://protsyk.com/cms/wp-content/uploads/2011/12/QuadCircle_1.png [Broken]
 
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  • #7
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
jaja1990 said:
"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! :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:
 
  • #9
I understand now, thank you for bearing with me! :biggrin:
 

1. What is the definition of area?

The area is the measure of the size of a surface or a two-dimensional shape. It is expressed in square units such as square meters or square feet.

2. How do you find the area of a shape?

The formula for finding the area of a shape depends on the shape itself. For example, the area of a rectangle can be found by multiplying its length by its width. The area of a circle can be found by using the formula A = πr², where r is the radius of the circle.

3. Why is finding the area important?

Finding the area of a shape allows us to quantify the amount of space it takes up. This is useful in many fields such as architecture, engineering, and mathematics. It also helps us to compare and analyze different shapes.

4. What are some real-life applications of finding the area?

Finding the area has many real-life applications. For example, it is used in designing buildings and determining the amount of material needed for construction. It is also used in farming to determine the amount of land needed for crops and in cartography to measure the size of a piece of land.

5. Can the area of a shape be negative?

No, the area of a shape cannot be negative. It is always a positive value because it is a measure of the space enclosed within the shape. If the shape has an irregular or concave shape, the area may be expressed as a negative value but it is still the same in terms of magnitude.

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