General Relationship Between Area & Perimeter

In summary, the question is asking if there is a general relationship between the area and perimeter of plane figures. While there is no set formula, there are general conclusions that can be made. 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), but it is not directly related to the question. For convex figures, there are better results.
  • #1
jason17349
22
0
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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  • #2
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.
 
  • #3
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).
 
  • #4
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.
 
  • #5
"
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.
 

1. What is the relationship between area and perimeter?

The area and perimeter of a shape are two important measurements related to its size. The area is the measure of the surface enclosed by the shape, while the perimeter is the distance around the shape's outer boundary. In general, the area and perimeter of a shape are inversely proportional, meaning that as one increases, the other decreases.

2. How are area and perimeter calculated?

The formula for calculating the area of a shape varies depending on the type of shape. For example, the area of a rectangle is calculated by multiplying its length by its width. The formula for perimeter also varies, but in most cases, it involves adding up the lengths of all sides of the shape. It is important to note that different shapes have their own specific formulas for calculating area and perimeter.

3. Is there a relationship between the area and perimeter of a square?

Yes, there is a relationship between the area and perimeter of a square. Since a square has equal sides, its area is calculated by multiplying the length of one side by itself. The perimeter of a square is calculated by multiplying the length of one side by 4. This means that as the length of one side increases, both the area and perimeter of the square also increase.

4. How does the relationship between area and perimeter apply to real-life situations?

The relationship between area and perimeter is seen in many real-life situations, such as when calculating the amount of fencing needed for a yard or the amount of paint needed to cover a wall. In these cases, as the area of the surface increases, so does the perimeter needed to enclose or cover it. This relationship is also important in fields such as architecture and engineering, where precise measurements of area and perimeter are crucial.

5. Can the relationship between area and perimeter be used to compare two shapes?

No, the relationship between area and perimeter cannot be used to directly compare two different shapes. While two shapes may have the same area, their perimeters can be completely different. For example, a circle and a rectangle can have the same area, but their perimeters will be different. Therefore, it is important to consider both the area and perimeter when comparing shapes.

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