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  ### Perimeter of 1000m
  
  ```
  Date: 07/13/99 at 14:59:31
  From: Louise Macdonald
  Subject: A shape with the maximum area and a perimeter of 1000m
  
  We have to find a shape with a perimeter of 1000m and the maximum
  area. I have tried to work this out but need to know if a circle
  would have the biggest area. I have already looked at rectangles,
  triangles, and polygons and have noticed that the equilateral shape
  had the maximum area of each group of shapes.
  
  Thank you.
  ```
  
  ```
  Date: 07/13/99 at 15:09:48
  From: Doctor Anthony
  Subject: Re: A shape with the maximum area and a perimeter of 1000m
  
  Consider various regular polygons with n = number of sides. You will
  have a constant perimeter (= 1000m) and you want an expression for the
  area in terms of n. When n = 3 you have an equilateral triangle with
  each side 333.33 metres.
  
  area = (1/2)333.33^2 sin(60) = 48112.52 m^2
  
  For a square of side 250 m
  
  area = 62500 m^2
  
  From here you require a general formula. If a polygon has n sides, the
  angle subtended by any side at the centre is 360/n. Suppose the
  polygon is inscribed in a circle of radius r; then the area of any
  triangle formed by radii from the centre to the edges of a side of
  the polygon is
  
  (1/2)r^2.sin(360/n)
  
  and the area of the whole polygon is (n/2)r^2.sin(360/n)
  
  where the perimeter is given by 2n.r.sin(180/n) = 1000
  
  n.r.sin(180/n) = 500
  
  So you have two formulae to work with:
  
  r = 500/[n.sin(180/n)] .....(1)
  
  and area
  
  A = (n/2)r^2.sin(360/n)  .......(2)
  
  Check with n = 3  r = 500/[3.sin(60)] = 192.4500897
  
  A = (3/2)(192.4500897)^2 sin(120) = 48112.52
  
  and this agrees with the value we found earlier.
  
  Combining the two formulae we get the area of a regular polygon of n
  sides having a perimeter of 1000m:
  
  (500)^2
  A =  -----------------
  n.tan(pi/n)
  
  which can be changed to    (500^2/n).cot(pi/n)
  
  = (500^2/n).tan(pi/2 - pi/n)
  
  or if you want to express it in degrees
  
  A = (500^2/n).tan(90 - 180/n)
  
  Plot a graph of A against n and show that A increases with an increase
  in n, but at a rate that decreases as n increases. The obvious limit
  is when you have n -> infinity, in which case you have a circle. Find
  the area of the circle. This will be the limit toward which A is
  
  - Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/
  ```
  Associated Topics:
  High School Analysis
  High School Euclidean/Plane Geometry
  High School Geometry
  
  Search the Dr. Math Library:
  
   Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words
  
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explanation: The document is a math question and answer exchange from the Dr. Math forum. The user asks about finding the shape with the maximum area given a perimeter of 1000 meters. The response from Dr. Math provides a detailed explanation and derivation of the formula for the area of regular polygons with a given perimeter, ultimately concluding that a circle would have the maximum area.
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  ### Perimeter of 1000m
  
  ```
  Date: 07/13/99 at 14:59:31
  From: Louise Macdonald
  Subject: A shape with the maximum area and a perimeter of 1000m
  
  We have to find a shape with a perimeter of 1000m and the maximum
  area. I have tried to work this out but need to know if a circle
  would have the biggest area. I have already looked at rectangles,
  triangles, and polygons and have noticed that the equilateral shape
  had the maximum area of each group of shapes.
  
  Thank you.
  ```
  
  ```
  Date: 07/13/99 at 15:09:48
  From: Doctor Anthony
  Subject: Re: A shape with the maximum area and a perimeter of 1000m
  
  Consider various regular polygons with n = number of sides. You will
  have a constant perimeter (= 1000m) and you want an expression for the
  area in terms of n. When n = 3 you have an equilateral triangle with
  each side 333.33 metres.
  
  area = (1/2)333.33^2 sin(60) = 48112.52 m^2
  
  For a square of side 250 m
  
  area = 62500 m^2
  
  From here you require a general formula. If a polygon has n sides, the
  angle subtended by any side at the centre is 360/n. Suppose the
  polygon is inscribed in a circle of radius r; then the area of any
  triangle formed by radii from the centre to the edges of a side of
  the polygon is
  
  (1/2)r^2.sin(360/n)
  
  and the area of the whole polygon is (n/2)r^2.sin(360/n)
  
  where the perimeter is given by 2n.r.sin(180/n) = 1000
  
  n.r.sin(180/n) = 500
  
  So you have two formulae to work with:
  
  r = 500/[n.sin(180/n)] .....(1)
  
  and area
  
  A = (n/2)r^2.sin(360/n)  .......(2)
  
  Check with n = 3  r = 500/[3.sin(60)] = 192.4500897
  
  A = (3/2)(192.4500897)^2 sin(120) = 48112.52
  
  and this agrees with the value we found earlier.
  
  Combining the two formulae we get the area of a regular polygon of n
  sides having a perimeter of 1000m:
  
  (500)^2
  A =  -----------------
  n.tan(pi/n)
  
  which can be changed to    (500^2/n).cot(pi/n)
  
  = (500^2/n).tan(pi/2 - pi/n)
  
  or if you want to express it in degrees
  
  A = (500^2/n).tan(90 - 180/n)
  
  Plot a graph of A against n and show that A increases with an increase
  in n, but at a rate that decreases as n increases. The obvious limit
  is when you have n -> infinity, in which case you have a circle. Find
  the area of the circle. This will be the limit toward which A is
  
  - Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/
  ```
  Associated Topics:
  High School Analysis
  High School Euclidean/Plane Geometry
  High School Geometry
  
  Search the Dr. Math Library:
  
   Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words
  
  Submit your own question to Dr. Math
  Math Forum Home || Math Library || Quick Reference || Math Forum Search
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a math question and answer exchange from the Dr. Math forum. The user asks about finding the shape with the maximum area given a perimeter of 1000 meters. The response from Dr. Math provides a detailed explanation and derivation of the formula for the area of regular polygons with a given perimeter, ultimately concluding that a circle would have the maximum area.
  
  Label: question-and-answer