Fourth grade Geometry: What is the definition of area?

mytori

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Hello,

This is an easy question for all of you here.

Would you consider both of these two shapes to have one single area each?

1) Draw two shapes with the same area and with the same length of circumference. Is this the correct answer? Or, is the first shape made out of two areas and cannot be counted as a single area.

2) What is the definition of area?

area.png

I hope you would like to answer my super simple question in this advanced math forum.
 

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Well, actually, in doing some research to answer this question I found it's not quite as simple as it might seem, and it all comes down to how you define terms. When many people think about the word "shape," things like circles, rectangles, triangles, etc. spring to mind. However, there's a subcategory of shapes, called polygons. The definition of polygon that I learned was an single enclosed area bounded only by straight lines. Accordingly, a circle is a shape, but it's not a polygon because it must be drawn using curved lines. However, I literally just learned today that there's something called "complex polygons" which are allowed to intersect itself and form multiple "areas." So, for your problem, you can see that depending on exactly how terms are defined, the left figure you've drawn might be considered a single complex polygon and thus one shape, or it might be considered two adjacent squares and thus two shapes. Regardless of if the left figure is one shape or two shapes, you didn't include any measurements or units, so I can't say if they really have the same area and perimeter or not.

You might find these two pages from the website Math is Fun to be helpful. What is Area? and Polygonshttps://www.mathsisfun.com/geometry/polygons.html
 
Would you consider both of these two shapes to have one single area each?

View attachment 7668
What do you mean, precisely, by "one single area"? Are you referring to the numerical value of the enclosed regions? Or the number of regions?

1) Draw two shapes with the same area and with the same length of circumference. Is this the correct answer? Or, is the first shape made out of two areas and cannot be counted as a single area.
Is the first sentence the exercise, and the others are your responses? Or are they all part of the exercise?

2) What is the definition of area?
What does your textbook say?

Please be complete. Thank you! ;)
 
What is the defintion of area?

Well, actually, in doing some research to answer this question I found it's not quite as simple as it might seem, and it all comes down to how you define terms. When many people think about the word "shape," things like circles, rectangles, triangles, etc. spring to mind. However, there's a subcategory of shapes, called polygons. The definition of polygon that I learned was an single enclosed area bounded only by straight lines. Accordingly, a circle is a shape, but it's not a polygon because it must be drawn using curved lines. However, I literally just learned today that there's something called "complex polygons" which are allowed to intersect itself and form multiple "areas." So, for your problem, you can see that depending on exactly how terms are defined, the left figure you've drawn might be considered a single complex polygon and thus one shape, or it might be considered two adjacent squares and thus two shapes. Regardless of if the left figure is one shape or two shapes, you didn't include any measurements or units, so I can't say if they really have the same area and perimeter or not.

You might find these two pages from the website Math is Fun to be helpful. What is Area? and Polygons


Thank you for your reply!

Well, you understood my problem before me. Cause I see now that my question is incorrect.

The question is about making two shapes with the same circumference and area. Yes the shapes has the correct measurements. I guess the problem is more about the definition about one single shape. The children has so far only worked with areas -> rectangles and squares. The book only present shapes with centimeter cubes where one side is attached to another side. Like this:
area.png
So if the child draws a shape like the one on the left, would that be a correct answer? Because the question does not define the meaning of shapes. Or should only shapes similar to the book be correct? It is difficult because the child had this question in a test and did not get points for the shape to the left. I am just thinking how precise you have to be in a test question for 10 year olds. If the question has a lot of information the actual question can be hard to understand instead. The children has not worked with polygones yet...well yes they have but they don't know the meaning of polygones yet. Usually they start with that in seventh grade.

Thank you for your links.
 
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What do you mean, precisely, by "one single area"? Are you referring to the numerical value of the enclosed regions? Or the number of regions?


Is the first sentence the exercise, and the others are your responses? Or are they all part of the exercise?


What does your textbook say?

Please be complete. Thank you! ;)


Thank you for your answer!

Well, yes I was not clear myself about the question as you can see in my other reply. Sorry for that.

Yes the first sentence is the question to the exercise and the other two sentences are my thoughts.

My mother tongue is Swedish so that's why my English is a bit poor ;)
 
This was an interesting question :) Thanks for sharing and thanks to the people who answered it :)
 
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