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.
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.