Chapter 2

Close Bounds to the Ratio of the Circumference of a Circle to its Diameter and to the Ratio of a Circle's Area to the Square of its Radius

Such bounds will enable estimation of the error that may exist when determining either circumference from a given diameter or area from a given radius.  To determine these bounds, consider the 90o circle segment shown next.

To proceed, underestimate the circumference by summing the short chords as seen on the circle segment and then overestimate the circumference by summing the longer tangent chords.   If a very small step size is chosen for the calculations these two bounds should lie very close together.

Note that the ratio of the length of the longer tangent chord to the length of the short chord is the same as the ratio of the length OD to the length OC. 

The length of line OD is the radius r as are the lengths OA and OB, where r = √2.

Let the length of FB be ∆x and the length of AF be ∆y.  Let the coordinates of point A be (x,y). Then the coordinates of point C are: ( (x+∆x)/2, (y- ∆y)/2). 

AB is the hypotenuse of the rt triangle AOB.  The length of the short chord, AB, is:
AB = ∆x 2 + y2

From the coordinates of C find the length OC as:

OC = (x+∆x/2)2 + (y - ∆y/2)2

For our circle with radius √2 the ratio OD/OC is just √2 /OC.  This is the ratio of the length of the tangent chord to the length of the short chord.

Moving to the ratio of a circle's area to the square of its radius, a lower bound on this ratio will be found by summing the areas of the inner isosceles triangles AOB, and an upper bound will be found by summing the areas of the outer isosceles triangles that are based on the, long, tangent chord.  (For an isosceles triangle, its area is given by multiplying its height by half the length of its base.)

To assure close bounds, 10,000 steps in x, 0 <=  x <= 1, were employed in the top 45o segment of the circle for the calculations.  See sample cells of the spread sheet, shown using scientific notation, in the table following:

The expressions in row 61 can be viewed by clicking the (column, row) addresses that follow.

. .
N61 O61 P61 Q61 R61 S61 T61 U61 V61 W61
The upper and lower bounds provided by the chords are closer together than those for the areas. The sought ratio lies between 3.14159265275... and 3.14159265525....  The exact value is known as Pi with the symbol π. 

Pi is, as are most numbers, a transcendental number. See Wikipedia for more information on transcendentals.

Pi requires an indefinitely large number of numeric digits for precise representation.  Its calculation has been a challenge to many and more than 1012 of its most significant digits have now been found.

A published approximation is: π =3.141592653589793 which is seen to lie nicely within our calculated bounds.  Using such a value for the ratios, the errors in calculating a circle's:

            circumference = π * d, and area = π * r2,

given a circle's diameter or radius, will be quite small.

The circumference C of a circle is equivalently represented as:

         C =  2 * π * r  and for the unit circle,  C = 2 * π. 

For this and other reasons, angle is often expressed in units of the path length around the unit circle. Such angles are termed radians with 2π radians equivalent to 360o.  Neatly, the internal angles of a plane triangle sum to π radians.

As an exercise, the reader should consider how to calculate a table of sines or cosines, in small equal steps of angle, and construct it using a spreadsheet.


Numerical Optimization is introduced in conjunction with a design problem.

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