Comments on: What 2d shape has the maximum area for a given perimeter? http://warrenpiece.com/2d/what-2d-shape-has-the-maximum-area-for-a-given-perimeter Sat, 19 May 2012 21:03:17 +0000 http://wordpress.org/?v=2.7 hourly 1 By: sshanbhag_2000 http://warrenpiece.com/2d/what-2d-shape-has-the-maximum-area-for-a-given-perimeter/comment-page-1#comment-5427 sshanbhag_2000 Mon, 04 Jan 2010 14:26:59 +0000 http://warrenpiece.com/2d/what-2d-shape-has-the-maximum-area-for-a-given-perimeter#comment-5427 Try this method The logic here is to establish some trends (which can be verified) and then extrapolate the result. 1) Obviously when we say different shapes we means polygons or move from a triangle (min side closed figure) to a circle (polygon with n = infinity) 2) We also need to check whether a regular polygon (as compared with a polygon of unequal sides) gives the max value of Area/perimeter. No reason to believe it won’t. This should repeat for every polygon (different values on n) As a simple check in this below formula for area of triangle Take s = semi perimeter = 12, and a=5, b=4, c=3 We get an area = 6 units For same value of semi perimeter = 6, taken above you can take a= b=c = 4 units (equilateral) Now Area = 4 sqrt 3 > 6 Hence we have max value for Area/perimeter at regular (equal side case) polygon 3) We should safely assume that the ratio of Area/perimeter when plotted for different values of n (n =3 till infinity) should either increase or decrease for increasing value of n (without a max or min for any intermediate value or for 3<n<infinity) The site below gives a relation which can help a bit. It establishes the relationship of Area of polygon with radius and radius is related to angle for polygon (angle in turn is related to no of sides of a regular polygon) http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_AreaPerimeterRegularPolygons.xml The area of a regular polygon of n sides and radius r is. side = 2rsin (180/n) and n * (side) = perimeter Using the above relationships we should have a trend for increasing value on n 4) We can then also check for an equilateral triangle and a circle THIS WORK HAS BEEN DONE BY OTHER ANSWERERS AND HAS FOUND THE CIRCLE TO HAVE A LARGE AREA/PERIMETER RATIO Now since area of regular polygon is max at equal side case for triangle and assumed true for each polygon and as n increases the ratio of area/perimeter increases we should assume that circle has the largest ratio<br><b>References : </b><br> Try this method
The logic here is to establish some trends (which can be verified) and then extrapolate the result.
1) Obviously when we say different shapes we means polygons or move from a triangle (min side closed figure) to a circle (polygon with n = infinity)
2) We also need to check whether a regular polygon (as compared with a polygon of unequal sides) gives the max value of Area/perimeter. No reason to believe it won’t. This should repeat for every polygon (different values on n)

As a simple check in this below formula for area of triangle

Take s = semi perimeter = 12, and a=5, b=4, c=3
We get an area = 6 units

For same value of semi perimeter = 6, taken above you can take a= b=c = 4 units (equilateral)
Now Area = 4 sqrt 3 > 6
Hence we have max value for Area/perimeter at regular (equal side case) polygon

3) We should safely assume that the ratio of Area/perimeter when plotted for different values of n (n =3 till infinity) should either increase or decrease for increasing value of n (without a max or min for any intermediate value or for 3<n<infinity)

The site below gives a relation which can help a bit. It establishes the relationship of Area of polygon with radius and radius is related to angle for polygon (angle in turn is related to no of sides of a regular polygon)

http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_AreaPerimeterRegularPolygons.xml
The area of a regular polygon of n sides and radius r is.

side = 2rsin (180/n) and n * (side) = perimeter
Using the above relationships we should have a trend for increasing value on n

4) We can then also check for an equilateral triangle and a circle

THIS WORK HAS BEEN DONE BY OTHER ANSWERERS AND HAS FOUND THE CIRCLE TO HAVE A LARGE AREA/PERIMETER RATIO

Now since area of regular polygon is max at equal side case for triangle and assumed true for each polygon and as n increases the ratio of area/perimeter increases we should assume that circle has the largest ratio
References :

]]> By: KevinM http://warrenpiece.com/2d/what-2d-shape-has-the-maximum-area-for-a-given-perimeter/comment-page-1#comment-5426 KevinM Mon, 04 Jan 2010 13:37:59 +0000 http://warrenpiece.com/2d/what-2d-shape-has-the-maximum-area-for-a-given-perimeter#comment-5426 A circle will give you the largest area. Proving this for any possible shape is a bit complicated... basically you make an equation for a loop, and you can show that the maximum value for the area inside that loop will only be reached when the loop is a circle. So if it isn't a circle, you can add a little bit of area by rounding it out a little bit with the same perimeter.<br><b>References : </b><br> A circle will give you the largest area. Proving this for any possible shape is a bit complicated… basically you make an equation for a loop, and you can show that the maximum value for the area inside that loop will only be reached when the loop is a circle. So if it isn’t a circle, you can add a little bit of area by rounding it out a little bit with the same perimeter.
References :

]]> By: Geezah http://warrenpiece.com/2d/what-2d-shape-has-the-maximum-area-for-a-given-perimeter/comment-page-1#comment-5425 Geezah Mon, 04 Jan 2010 12:47:59 +0000 http://warrenpiece.com/2d/what-2d-shape-has-the-maximum-area-for-a-given-perimeter#comment-5425 A circle gives you the most area for its perimeter (circumference). This is part of a more general problem called the Isoperimetric inequality: http://en.wikipedia.org/wiki/Isoperimetric_inequality Basically, you can mathematically prove that the closed plane curve of a fixed length that gives you the biggest area, is the curve where each point is the same distance from a fixed point, which is a circle.<br><b>References : </b><br>BS & MS in mathematics A circle gives you the most area for its perimeter (circumference).

This is part of a more general problem called the Isoperimetric inequality:
http://en.wikipedia.org/wiki/Isoperimetric_inequality

Basically, you can mathematically prove that the closed plane curve of a fixed length that gives you the biggest area, is the curve where each point is the same distance from a fixed point, which is a circle.
References :
BS & MS in mathematics

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