General Relationship Between Area & Perimeter

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SUMMARY

The discussion centers on the relationship between area and perimeter in plane figures, specifically addressing regular and irregular polygons. It concludes that while there is no universal formula linking area and perimeter, general principles apply: for rectangles, a shape closer to a square maximizes area for a given perimeter. Additionally, Stoke's theorem is mentioned, which relates integration of forms over a manifold's interior and boundary, although it does not directly answer the original question regarding polygons. The key takeaway is that the configuration of a figure significantly impacts its area and perimeter.

PREREQUISITES
  • Understanding of basic geometric principles, including area and perimeter calculations.
  • Familiarity with properties of regular and irregular polygons.
  • Knowledge of Stoke's theorem and its application in geometry.
  • Concept of convex shapes and their properties.
NEXT STEPS
  • Research the properties of convex polygons and their area-perimeter relationships.
  • Study Stoke's theorem in detail and its implications in higher-dimensional geometry.
  • Explore optimization techniques for maximizing area given a fixed perimeter.
  • Investigate the relationship between shape configuration and geometric properties in irregular polygons.
USEFUL FOR

Mathematicians, geometry enthusiasts, educators, and students seeking to deepen their understanding of the relationships between area and perimeter in various geometric figures.

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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