Perimeter and Area of Regular Polygons - From Triangles to Circles - The Iso-Perimetric Theorem

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The Iso-Perimetric Theorem (IP Theorem; iso - same perimeter) concerns with the relation between the perimeter and the area of two dimensional (2-D) surfaces - plane figures.  For a historic account click here. A rigorous proof of IP Theorem is complex (see also) - I shall present here proofs that only need school level mathematics. 

IP Theorem may be stated as follows:

a. Among all 2-D shapes with the same perimeter, a circle has the largest area; or equivalently - 
Among all 2-D shapes with the same area, a circle has the shortest perimeter.

A short 2 minute video on youtube has a demonstration of the IP theorem.

For polygons (2-D shapes with n sides; n ≥ 3), IP Theorem may be stated as:

b.  For an n-sided polygon (fixed n; n-gon) with the same perimeter, the regular polygon (all sides of the same length) has the largest area.
c.  Among all regular polygons (any n) with the same perimeter, the one with the largest number of sides (largest n) has the largest area.
A circle may be considered a regular polygon approaching infinite number of sides and has the largest area for a given perimeter (points b and c). Statement c is thus equivalent to statement a of the theorem. 

A corollary of the IP Theorem may be stated as

d.  For a regular polygon for arbitrary n, the ratio of its area to its perimeter is equal to half the radius of the inner circle (half its apothem).

About Polygons:  The following two slides list the properties of regular polygons.  They relate the area and perimeter of n-sided polygons to their basic parameters like side length, circumradius and the apothem.  Centre of a regular poygon is the point that is equidistant from all vertices of the polygon - it is the centre of the circumcircle.

(Click on a slide to see full page view, Escape to return to text)
Slide 1

Slide 2

Eq.7 gives us the ratio of the area A to the perimeter of a regular polygon.  For a given parameter P, the maximum value the ratio (A/P) may have, is r/2 and that is for a circle for which n tends to infinity and cos (𝜋/n) = 1. For all other values of n, area A of the polygon is less than Pr/2. I show this in the next slide


Slide 3

We can make an interesting observation, from the graph on this slide:  The ratio of the area and the perimeter of a regular polygon is equal to half the length of the inner circle radius (the apothem). This result is independent of the number of sides in the polygon. Length of the apothem y does increase with n according to eqns. 1 and 2; namely y = r cos(𝜋/n).


Slide 4

We also notice that the circle has the largest area for a given perimeter (A/P approaches 0.5 r as n tends to infinity).
In the above, we have proved statements a, c and d of the isoperimetric theorem. Next let us look at statement b.

Regular n-gons have the largest area for a given perimeter:  To understand this, we use a heuristic approach and look at triangles formed on a chord in a circle with the vertices of the triangles on the circumference. See the following two slides:


Slide 5

Slide 6

 The Equilateral Triangle:  We shall use algebra to show that 
for a triangle (n = 3), for a given perimeter an equilateral triangle has the largest area.  
I shall use the well known Heron's formula for area of a triangle.

Slide 7

Slide 8

The Square:  First we look at the general four-sided polygon (a quadrilateral) and show that the rectangle (all internal angles equal to 90 degrees) has the largest area for a given perimeter.  Then we use the 
 AM-GM theorem (the arithmetic mean of a set of numbers is always greater than or equal to their geometric mean) to show that of all rectangles, a square (a regular 4-gon; all sides are equal and angles = 90 degrees) has the largest area.
Here, I have followed the analysis by Martin Joseffson in Geometricorum 13, 2013, 17-21

Slide 9

Slide 10

An interesting historical remark is that the formula K = (a+c)/2 x (b+d)/2 was used by the ancient Egyptians to calculate the area of a quadrilateral, but it’s only a good approximation if the angles of the quadrilateral are close to being right angles. In all quadrilaterals but rectangles the formula gives an overestimate of the area, which the tax collectors probably didn’t mind!

AM-GM Theorem: I now show that the largest area for a rectangle is for a square when all four sides are equal. 
Slide 11

Notes:  
1.  I have considered convex polygons in the discussion.  For completeness, I present a slide to explain the difference between a convex and a concave polygon.

Slide 12

2.  I find the case when y = 2 very interesting and have prepared a slide to show the numbers for some polygons. Please refer to slides 1 and 2 for notations.
Slide 13

In a circle, the inner and circumradius are always the same.  

3. I have drawn all the slides in this blog.  You can use them but acknowledge the source of this blog as 
http://ektalks.blogspot.com/2018/09/perimeter-and-area-of-regular-polygons.html