General Relationship Between Area & Perimeter

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Discussion Overview

The discussion centers on the relationship between area and perimeter in plane figures, including both regular and irregular polygons. Participants explore whether a general formula exists that connects these two properties across different shapes.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions if there is a general relationship between area and perimeter for various plane figures, including irregular polygons.
  • Another participant states that there is no set formula for determining area based on perimeter, but notes that general conclusions can be drawn, particularly for rectangular figures.
  • It is mentioned that for rectangles, the area is maximized when the shape is a square, while a rectangle with the same perimeter can have varying areas depending on its dimensions.
  • A participant introduces Stoke's theorem, suggesting a relation between integration over the interior and boundary of a manifold, although this is noted as not directly related to the original question.
  • Another participant asserts that while there is a maximum area that can be enclosed by a given perimeter in the plane, there is no minimum area, particularly when considering convex figures.
  • A later reply questions the relevance of the Stoke's theorem reference made earlier, indicating some uncertainty about its applicability to the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a general relationship between area and perimeter. Some agree that while no universal formula exists, certain observations can be made, while others emphasize the limitations of these observations, particularly in relation to irregular shapes.

Contextual Notes

The discussion highlights the complexity of defining relationships between area and perimeter, especially for irregular polygons, and the assumptions made regarding the shapes being considered (e.g., convexity).

Who May Find This Useful

This discussion may be of interest to those studying geometry, particularly in understanding the properties of shapes and the relationships between their dimensions.

jason17349
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This is kind of a vague question but does anybody know if there is a more general relationship between the area and perimeter of plane figures. For example circles, squares, rectangles triangles any regular polygon really, the area can be written in terms of the perimeter. Is there anything that can extended this idea to irregular polygons? Thanks.
 
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There is no set formula for determining the area of a figure based upon its perimeter or visa versa. However there are general conclusions that can be made based on either factor (area/perimeter). For instance, a rectangle with a perimeter of 24 units can have an area of 36 square units if it is a perfect square. A figure with the same perimeter of 24 units could have an area of 11 square units given the fact that its dimensions are 1 x 11. Generally speaking, for rectangular figures, the closer it is to being a perfect square, the greater its area. The greater the difference between length and width of the figure, the greater the perimeter. Just remember that it is based upon the chosen method of determining size. If the figure is defined by its area, then it will have the greatest area in the form of a square. It will have the greatest perimeter in a 1 x __ rectangle. If the figure is determined by perimeter, then it will have the greatest perimeter in the form of a 1 x__ rectangle. It will have the greatest area in the form of a square. This may seem a little wordy, but I want to be thorough.
 
Not directly related to your question but still...

Stoke's theorem gives a relation between integration of a k-form over the interiors and its (k+1)-form over the boundary of the same structure (manifold).
 
If you're working in the plane, there's a largest area that can be enclosed by a set perimeter. However, there is no smallest area that can be enclosed by a set perimeter. If you require that your figures are convex, there are better results.

In short, the answer is no.
 
"
guhan said:
Not directly related to your question but still...

Stoke's theorem gives a relation between integration of a k-form over the interiors and its (k+1)-form over the boundary of the same structure (manifold).
"

Are you sure guhan from tambaram.
 

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