Pi is an interesting number. It is defined as the ratio of a circle’s circumference to its diameter, which ancient civilisations realized to be a constant. Given a circle with a diameter D or radius r and circumference C, mathematically, it is expressed as pi = C/D or pi = C/2r.

We’ve come across different numbers such as 22/7, 3.14, 3.1416 as the value of pi in solving mathematical problems relating to circle or circular shape. In other words, pi appears to be not having an exact numeric value. Is it a rational number that can be expressed as a fraction? How can we determine its value?

Calculating the value of pi

There are many different methods to determine the approximate value of pi. The simplest way to get the answer is by applying Pythagorean theorem (i.e. c^{2} = a^{2} + b^{2}) and properties of isosceles triangle, which the Greek mathematician Archimedes had came up with.

Consider a circle with diameter D as shown in Figure 1 below. A regular hexagon “ABCDEF” can be constructed inside the circle with six equilateral triangles (i.e. ∆OAB, ∆OBC, ∆OCD, ∆ODE, ∆OEF, ∆OFA) with each side equal to the radius r.

The perimeter of the hexagon will be approximately equal to the circumference. If the hexagon is further sub-divided into a 12-sided polygon (see Figure 2 above). The shape of the 12-sided polygon “AGBHCJDKELFM” will be more like a circle. In other words, when an n-sided polygon is drawn inside the circle, the polygon shape will be almost like a circle, but it is not a true circle. The perimeter of an n-sided polygon will approach the circumference, hence an accurate value of pi can be obtained. The value obtained by this method must be less than the exact value of pi. It is never the exact value of pi.

Step 1

Let’s consider the circle in Figure 1 with radius r equal to 1 (i.e. D = 2). For the equilateral triangle ∆OAB (see Figure 3), OA = OB = AB = 1.

The perimeter of the hexagon “ABCDEF” = 6 x AB = 6. Applying the formula mentioned above, pi = C/D = 6/2 = 3.

Step 2

Consider the equilateral triangle ∆OAB in Figure 4. It is also an isosceles triangle with OA = OB. A perpendicular bisector OP is drawn from the vertex O towards AB to form an intersection point P. Therefore /OPA = /OPB = 90^{o}, and AP = PB [Properties of isosceles triangle]. The perpendicular bisector OP is extended to meet the circle at point Q, where OQ = 1, /APQ = /BPQ = 90^{o}[Opposite angle].

When each equilateral triangle inside the circle in Figure 1 is subdivided to form two isosceles triangles, a dodecagon (12-sided polygon) “AGBHCJDKELFM” is formed as in Figure 2. In other words, the dodecagon is made up of 12 equal isosceles triangles.

Perimeter of the 12-sided polygon = 12 x 0.517638 = 6.211657 An approximation of pi = C/D = 6.211657/2 = 3.105829 Value of pi is now closer to our known practical value

Below is the output of the spreadsheet for calculation of pi by using a 12-sided polygon and applying the formulae in Figure 5.

Step 3

For the triangle ∆OAQ in Step 2 (see Figure 6 below), it is also isosceles with OA = OQ. A perpendicular bisector OR is drawn from the vertex O towards AQ to form an intersection point R. Therefore /ORA = /ORQ = 90^{o}, and AR = RQ. The perpendicular bisector OR is extended to meet the circle at point S, where OS = 1, /ARS = /QRS = 90^{o}.

Two isosceles triangles (i.e. ∆OAS and ∆OSQ) are obtained. The other isosceles triangles in the 12-sided polygon follow the same way of division. Hence 24 isosceles triangles with the same shape as ∆OAS are evenly distributed inside the circle to form a 24-sided polygon. Its perimeter is closer to the circumference as compared with the 12-sided polygon.

Perimeter of the 24-sided polygon = 24 x 0.261052 = 6.265257 An approximation of pi = C/D = 6.265257/2 = 3.132629 12-sided polygon gives an approximation of pi = 3.105829 24-sided polygon gives an approximation of pi = 3.132629

Below is the output of the spreadsheet for calculation of pi by using a 24-sided polygon and applying the formulae in Figure 6.

Step 4

Following the same methodology above to sub-divide the isosceles triangles of the 24-sided polygon to form a 48-sided one, it gives an approximation of pi = 3.139350 (see output of the calculation spreadsheet below).

The above steps illustrate dividing of the hexagon repeatedly to form an n-sided polygon and its perimeter can be obtained by applying the Pythagorean theorem to the sub-divided isosceles triangles.

Below is the spreadsheet for genearating the approximate value of pi from polygons with the number of sides increasing from 48 up to 6291456.

From the above, the value of pi is increasing from 3 for a hexagon and steadily up to 3.141592653589660 for a 6291456-sided polygon, but it won’t stop there because the number of sides of the polygon could be infinite. pi cannot be a rational number nor represented by a fraction. Practically 3.14159 is chosen as it could give a quite accurate result in solving most mathematical and engineering problems.

As shown above, the value of pi determined by this method reveals that it must be less than the exact value of pi. Is there any method to evaluate the upper boundary of pi? If yes, it would give us the range that pi lies within.

Boundaries of the value of pi

Let’s consider a regular hexagon A’B’C’D’E’F’ circumscribing a circle as shown in Figure 7.

So, there is a relationship between the inscribed and circumscribed regular polygons. Let’s consider a regular hexagon where AP = 0.5, OP = 0.866025 [from Step 2], hence A’Q = 0.5/0.866025 = 0.577350, perimeter of the circumscribed hexagon A’B’C’D’E’F’ is 6 x 2 x 0.577350 = 6.928203, pi is approximately equal to 6.928203/2 = 3.464102.

Following the same process, the 12-sided regular circumscribed polygon gives pi = 3.215390.

Similarly, the 24-sided regular circumscribed polygon gives pi = 3.159660.

Below is the spreadsheet for genearating the approximate value of pi from inscribed and circumscribed polygons with the number of sides increasing from 48 up to 6291456.

From the analysis of circumscribed polygons, the value of pi is decreasing gradually from 3.464102 for a hexagon to 3.141592653590050 for a 6291456-sided polygon. Based on the results of inscribed and circumscribed polygons, the value of pi lies between two boundaries:

Conclusion

There is no exact value of pi. It is an irrational number and cannot be expressed as a fraction. There is no repeating pattern after the decimal point.

The boundaries of pi are not fixed. They depend on the number of sides of the polygons inscribed in and circumscribed the circle used for calculation. With a polygon of larger number of sides adopted in the analysis, the narrower the range of the value of pi becomes.

It is reasonable to adopt 3.14159 for pi in solving most mathematical and engineering problems, which will give a quite accurate result. The methodology for analysis is just like the principles adopted in calculus.