Contents

On the approximation of: THE AREA OF A  REGULAR  POLYGON

On the approximation of: THE CIRCUMFERENCE OF AN ELLIPSE

THE BINODAL LEMNISCATE

A STRAIGHT LINE

On the approximation of: THE AREA OF A  REGULAR  POLYGON

The Journal of Recreational Mathematics, Vol.31(2) 81-82,2001 has an article on the approximation of the area of a regular polygon. There is sometimes a need to use the approximation when making an estimate, even though there may be an exact formula which uses trigonometry. It can almost be done in your head.

The approximation is not too good for the equilateral triangle. It is quite close for the square. But, for n=5, the pentagon and upwards the error is slight.

Let  “e“ be the  length of one edge and  n be  the number of edges, or  sides.  
And    π = 22/7,  if you wish, but more accurately:

            π = 3.1415926535897932384626433 approximately

The approximation is intended for the ordinary citizen and Middle School, or Junior High School student who might like to find the area or for anyone who wants an estimate.

 The approximation is given by:

A(n,e) = (n² – π)e²/4π              ….(1)

No special trigonometry tables are required, Using 22/7, or a better approximation of π, makes little difference. You will still end up with an error of about 1% on the high side.

Since we know the error is always on the high side by about 1%, a second estimate would be one percent lower, which is easy to calculate

 EXAMPLE

 Brian the gardener wanted to estimate the area of a pentagon flower bed he had in mind. It would be 2 meters along each edge and a regular pentagon.

Using  π=22/7, n=5, e=2 meters in Equation 1, we have:

 A(n,e) = (n² – π)e²/4π     
           =  (25 – 22/7)2²/(88/7)
           = 153Χ4/88
           = 153/22
           =6.95454545…

 less    .069545454 which is one percent

      =6.88 square meters  which is the same as  the exact formula by trigonometry.

On the approximation of the  circumference of an ellipse

 An approximation to the circumference of an ellipse was shown in the newsletter of the Statistical Association of Manitoba, Vol. 14, No. 5,  January 1991. The approximation is closer than the error induced by using 22/7 for the value of p. The equation is:

S = 2(p - (p -2).e³).a
where: 
p = 3.1415926535897932384626433832795  approximately.
e   is the eccentricity, and   0 £ e < 1
a     is the length of the semi-major axis.

THE BINODAL LEMNISCATE

It can be shown that Cassini’s Lemniscate is derived out of the equations of  two circles:

(x-a)² + y² = k₁²                                   …(1)

(x+a)² + y² = k₂²                                      (2)

which when multiplied together produce:

{(x-a)²+y²}{(x+a)²+y²)}=k₁²k₂²=b⁴     …(3)

 and, depending upon the value of b become ovals, lemniscate, or ellipse. Now, in order to obtain another loop, or node, bring in another circle:

 x² + y² = k₃²                                         …(4)

and obtain,

 {(x-a)²+y²}(x²+y²){(x+a)²+y²)}=k       …(5)

 where the new k is to be determined. Set y=0 allows us to obtain and obtain where the curve intercepts the x-axis.

 (x-a)²x²(x+a)² = x²(x²-a²)² = k            …(6)

 or,

 x⁶ – 2a²x⁴ + a⁴x² – k = 0                     …(7)

In order to have the intercepts at x=±A, and  x=±B one also requires:

(x-B)(x-A)²(x+A)²(x+B) = 0               …(8)       

 Both Equations 7 and 8 represent the  curve from different points of view.  The coefficients must be equal, so  it requires solving the simultaneous equations:

                                                 2a² = 2A² + B²                         …(11)

a⁴ = A⁴+2A²B²                     …(12)

k  = A⁴B²                          …(13)

With effort, you will find:

B² = 4a²/3                      …(14)

a²=3A²                           …(15)

k=4a⁶/27                        …(16)

The nodes appear at: (±2a/√3, 0) and the x-intercepts are found at (±2a/√3, 0).

A STRAIGHT LINE

Excerpted from:
 John R. Hendricks, Journal of Recreational Mathematics, 14:2, 1982,pp.86-88

The article starts off by stating:
A straight line is a line without kinks, wiggles, or bends.

It then discusses;

1. Shortest distance

2. The Segment Approach

3. A Simple Definition
   A straight line is the path of a point moving constantly in the same direction.

4. Definition by Equation

5. Degrees of Freedom

6. Conclusion;
   The definition at the beginning of this article is probably as good as one can get until something better comes along.
   On the other hand, maybe everyone knows what a straight line is and so it requires no definition.

Last updated Thursday March 22, 2007

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