Pi Calculations

Pi (π) is defined as the Circumpherence (C) of a circle divided by its diameter (d).

Table of Contents

1. Using Trigonometry
2. Using Differential Calculus
3. Using Integral Calculus
4. Using Polygons
5. Precision

1. Using Trigonometry

Key:

C	Circumference
Ci	Circumference, inner polygon
d	diameter
r	radius
xi	point x-inner value
yi	point y-inner value
θ	theta, angle at the center
ψ	psi, angle at the periphery near the circle

Method

Take a generic polygon, in this case a hexagon (6 sides), and divide it as shown.

Notice that you can divide the triangle to the upper right and create a length of y_i.

In this case, you can use y_i as a rough estimate of the radius (r).

By taking ever smaller values of angle (π), and ever smaller values of y_i, then multiply inversely proportional to the angle to get a rough estimate of the circumpherence (C), you can then divide this by the diameter (d) to get pi (π).

Formulae

sin θ = $\frac{\mathrm{y\; _i}}{\mathrm{r}}$

Since r = 1:

sin θ = y _i

π = $\frac{\mathrm{C}}{\mathrm{d}}$ = $\frac{\mathrm{C}}{\mathrm{2r}}$

Calculations

θ = 60° = 1.047 radians

sin(1.047) = y_i = 0.866

6 y_i = 6 x 0.866 = 5.196 < C

π = $\frac{\mathrm{C}}{\mathrm{2r}}$ = $\frac{5.196}{2}$ = 2.598076211

Note: this is only a first approximation. If you double the divisions from 6 to 12 to 24, and so on, you can build a table of calculations that follows. As you approach 18 iterations, or 786,432 divisions, you approximate π by six decimal points. You can carry out his calculation to get as much precision as necessary. π is an irrational number, so that the series repeats to infinity.

Table

#	Div	Deg	Rads	yi	mult	π (÷2)
1	6	60	1.047	0.866	5.196	2.598076211
2	12	30	0.524	0.500	6.000	3.000000000
3	24	15	0.262	0.259	6.212	3.105828541
4	48	7.5	0.131	0.131	6.265	3.132628613
5	96	3.75	0.065	0.065	6.279	3.139350203
6	192	1.875	0.033	0.033	6.282	3.141031951
7	384	0.9375	0.016	0.016	6.283	3.141452472
8	768	0.46875	0.008	0.008	6.283	3.141557608
9	1,536	0.234375	0.004	0.004	6.283	3.141583892
10	3,072	117.19 x10-3	0.002	0.002	6.283	3.141590463
11	6,144	58.59 x10-3	0.001	0.001	6.283	3.141592106
12	12,288	29.30 x10-3	0.001	0.001	6.283	3.141592517
13	24,576	14.65 x10-3	0.000	0.000	6.283	3.141592619
14	49,152	7.32 x10-3	0.000	0.000	6.283	3.141592645
15	98,304	3.66 x10-3	0.000	0.000	6.283	3.141592651
16	196,608	1.83 x10-3	0.000	0.000	6.283	3.141592653
17	393,216	915.53 x10-6	0.000	0.000	6.283	3.141592653
18	786,432	457.76 x10-6	0.000	0.000	6.283	3.141592654
∞						3.141592654

Graph

---

2. Using Differential Calculus

Key:

C	Circumference
d	diameter
r	radius
xo	point x at the initial iteration, on the circle
yo	point y at the initial iteration, on the circle
xn	point x after n iterations
yn	point y after n iterations
θo	angle theta, at the initial iteration
θn	angle theta, after n interstions

Method

Take ever smaller values for theta (θ), using the differential equation listed below.

Formulae

sin θ = $\frac{\mathrm{y_o}}{\mathrm{r}}$

r = 1

Therefore, we simply to:

sin θ = y_o

Fron Trigonometry, we know that:

$\stackrel{\lim }{\theta \to 0}\frac{\mathrm{sin\; \theta }}{\mathrm{\theta }}$ → 1

Substitute π/n for θ:

$\stackrel{\lim }{\pi /n\to 0}\frac{\mathrm{sin\; \pi /n}}{\mathrm{\pi /n}}$ → 1

Simplify the expression on the left by inverting the ratio:

$\stackrel{\lim }{\pi /n\to 0}\frac{\mathrm{n}}{\mathrm{\pi }}$ [sin π/n] → 1

Multiply both sides of the expression by π:

$\stackrel{\lim }{\pi /n\to 0}$ n [sin π/n] → π

Calculations

First, let us take n = 6:

sin(π/n) = sin(π/6) = 0.5

n sin(π/n) = 6 x 0.5 = 3.0

Therefore, π must be greater than 3.0. If we take iterations of n by doubling the value 6 to 12, then 12 and so forth, we can create a table that follows.

Table

#	n	sin(π/n)	n sin(π/n)
1	6	0.500	3.000000
2	12	0.259	3.105829
3	24	0.131	3.132629
4	48	0.066	3.139350
5	96	0.033	3.141032
6	192	0.016	3.141452
7	384	0.008	3.141558
8	768	0.004	3.141584
9	1,536	0.002	3.141590
10	3,072	0.001	3.141592
11	6,144	0.0005	3.141593
12	12,288	0.0003	3.141593

Note: after 11 iterations, π has been calculated to six decimal points.

Graph

---

3. Using Integral Calculus

Key

C	Circumference
d	diameter
r	radius
A	Area, in this case half a circle

Formulae

C = 2 π r

A = π r²

½ A = $\frac{\mathrm{\pi \; r²}}{2}$

r = 1

½ A = $\frac{\mathrm{\pi }}{2}$

x² + y² = r²

Solve this as a function of x:

f(x) = sqrt(r² - x²)

f(x) = sqrt(1 - x²)

Take the integral from -1 to 1, which is ½ A:

$\frac{\mathrm{\pi }}{2}={\int }_{-1}^1\sqrt{\mathrm{1\; -\; x²}}dx$

π = 2 ${\int }_{-1}^1\sqrt{\mathrm{1\; -\; x²}}dx$

If you divide the area by 0.5 increments, and use the above formula to calculate the height of each rectangle, you shall have four areas to measure to produce the following table:

Table, divide the half circle by 4 rectangles

x	x-diff	f(x)	An
-1		0.000
-0.5	0.5	0.866	0.433
0	0.5	1.000	0.500
0.5	0.5	0.866	0.433
1	0.5	0.000	0.000
½π			1.366
π			2.732

Continue by creating 8 rectangles under the half circle:

Table, divide the half circle by 8 rectangles

x	x-diff	f(x)	An
-1
-0.75	0.25	0.661	0.165
-0.5	0.25	0.866	0.217
-0.25	0.25	0.968	0.242
0	0.25	1.000	0.250
0.25	0.25	0.968	0.242
0.5	0.25	0.866	0.217
0.75	0.25	0.661	0.165
1	0.25	0.000	0.000
½π			1.498
π			2.996

You will notice quickly that the value of π is increasing. You can continue the process repeatedly until you have has many digits of accuracy as you wish. In my case, I shall continue until I have 6 digits:

Table, iterating the area of a half circle, then multiply by 2

Continue by creating 16 rectangles:

Table, divide the half circle by 16 rectangles

x	x-diff	f(x)	An
-1
-0.875	0.125	0.484	0.061
-0.75	0.125	0.661	0.083
-0.625	0.125	0.781	0.098
-0.5	0.125	0.866	0.108
-0.375	0.125	0.927	0.116
-0.25	0.125	0.968	0.121
-0.125	0.125	0.992	0.124
0	0.125	1.000	0.125
0.125	0.125	0.992	0.124
0.25	0.125	0.968	0.121
0.375	0.125	0.927	0.116
0.5	0.125	0.866	0.108
0.625	0.125	0.781	0.098
0.75	0.125	0.661	0.083
0.875	0.125	0.484	0.061
1	0.125	0.000	0.000
½π			1.545
π			3.090

Continue dividing until you reach the desired number of digits:

Table, with divisions (Div) and the value of pi (π)

Div	π
4	2.732051
8	2.995709
16	3.089819
32	3.123253
64	3.135102
128	3.139297
256	3.140781
512	3.141306
1024	3.141491
2048	3.141557
4096	3.141580
∞	3.141593

Graph

 

---

4. Using Polygons

Key

C	Circumference
Ci	Circumference, inner polygon
Co	Circumference, outer polygon
d	diameter
r	radius
xi	point x-inner value
yi	point y-inner value
xo	point x-outer value
yo	point y-outer value
li	line, inner value, inside the circle
lo	line, outer value, outside the circle
θ	theta, angle at the center
θi	theta, inner angle at the center
θo	theta, outer angle at the center
ψ	psi, angle at the periphery near the circle
ψi	psi, inner angle at the periphery
ψo	psi, outer angle at the periphery

Method

Take a generic polygon, say the Pentagon, and define the inner line (l_i) and the outer line(l_o). Calculate each of these values and multiply by the number of divisions: in this case 5. The inner lines give the lower limit and the outer lines give the upper limit of π. Take higher orders of polygons to narrow the range.

Formulae

sin θ_i = y_i

sin ψ_i = $\frac{\mathrm{y_i}}{\mathrm{l_i}}$

sin θ_o = y_o

sin ψ_o = $\frac{\mathrm{y_o}}{\mathrm{½\; l_o}}$

r = 1

π = $\frac{\mathrm{C}}{\mathrm{d}}$ = $\frac{\mathrm{C}}{\mathrm{2r}}$

Calculations

Inner Pentagon

θ_{i = 72° = 1.257 rads}

sin θ_{i = yi}

sin (1.257) = y_i

y_i = 0.951

sin ψ_i = $\frac{\mathrm{y_i}}{\mathrm{l_i}}$

ψ_i = 54° = 0.942 rads

l_i = $\frac{\mathrm{y_i}}{\mathrm{sin\; \psi _i}}$ = $\frac{0.951}{\mathrm{sin(0.942)}}$ = 1.176

C_i = 5 l_i = 5 x 1.176 = 5.878

π_i = $\frac{\mathrm{C_i}}{\mathrm{2r}}$ = $\frac{5.878}{2}$ = 2.940

Outer Pentagon

θ_o = 36° = 0.628 rads

sin θ_o = y_o

sin (0.628) = y_o = 0.588

ψ_o = 54° = 0.942 rads

l_o = $\frac{\mathrm{2\; y_o}}{\mathrm{sin\; \psi _o}}$ = $\frac{\mathrm{2\; x\; 0.588}}{\mathrm{sin(0.942)}}$ = 1.453

C_o = 5 l_o = 5 x 1.453 = 7.265

π_o = $\frac{\mathrm{C_o}}{\mathrm{2r}}$ = $\frac{7.265}{2}$ = 3.633

Therefore:

π_i < π < π_o

2.939772 < π < 3.632713

Note: you can do the same calculations for a series of polygons and create the following tables:

Table of Angles used in the calculations

n	Polygon	θi	ψi	θo	ψo
3	Triangle	60		60
4	Square	45		45
5	Pentagon	72	54	36	54
6	Hexagon	60	60	30	60
7	Heptagon	51.43	64.29	25.71	64.29
8	Octagon	45	67.50	22.5	67.5
9	Nonagon	40	70	20	70
10	Decagon	36	72	18	72

Table of Calculations using Polygons

n	πi	range	πo
3	2.598076	< π <	5.196152
4	2.828427	< π <	4.000000
5	2.938926	< π <	3.632713
6	3.000000	< π <	3.464102
7	3.037186	< π <	3.371022
8	3.061467	< π <	3.313708
9	3.078181	< π <	3.275732
10	3.090170	< π <	3.249197

Graph

---

Special Case of the Triangle and Square

You will notice that the triangle and square represent special cases with regards to the polygons. The calculations are included here for completeness.

Triangle

Inner Triangle

θ = θ_i

θ_i = 60° = 1.047 rads

sin θ_i = y_i

sin(1.047) = y_i = 0.866

C_i = 6 x 0.866 = 5.196

π_i = $\frac{\mathrm{C_i}}{\mathrm{2r}}$ = $\frac{5.196}{2}$ = 2.598076

Outer Triangle

θ = θ_o

θ_o = 60° = 1.047 rads

sin θ_o = $\frac{\mathrm{y_o}}{\mathrm{r\; +\; l_v}}$

cos θ_o = $\frac{\mathrm{r}}{\mathrm{r\; +\; l_v}}$

Solving for (r+l_v):

r + l_v = $\frac{\mathrm{y_o}}{\mathrm{sin\; \theta _o}}$

r + l_v = $\frac{\mathrm{y_o}}{\mathrm{cos\; \theta _o}}$

Then, equating the two expressions gives:

y_o = $\frac{\mathrm{sin\; \theta _o}}{\mathrm{cos\; \theta _o}}$

y_o = $\frac{\mathrm{sin(1.047)}}{\mathrm{cos(1.047)}}$ = 1.732

C_o = 6 x 1.732 = 10.392

π_i = $\frac{\mathrm{C_o}}{\mathrm{2r}}$ = $\frac{10.392}{2}$ = 5.196152

Square

Inner Square

θ = 45° = 0.785 rads

sin θ = y_i

sin(0.785) = y_i = 0.707

C_i = 8 x y_i = 8 x 0.707 = 5.657

π_i = $\frac{\mathrm{C_i}}{\mathrm{2r}}$ = $\frac{5.657}{2}$ = 2.828427

Outer Square

y_o = r

C_o = 8 x r = 8 x 1 = 8

π_o = $\frac{\mathrm{C_i}}{\mathrm{2r}}$ = $\frac{8}{2}$ = 4.000000

---

5. Precision

I have calculated π to six decimal places: 3.141593. π to 9 decimal places is 3.141592654. The difference between these two values is 346 x10^-9, or a ratio of 1:2,890,173.412, or roughly 1 in 3 million! This begs the question, how precise do you need to be?

In 1873, William Shanks calculated π to 527 decimal points. According to Live Science and ScienceAlert, π has been calculated to 314 trillion digits using a Dell PowerEdge R7725 server. These are just exercises of endurance and prestige, respectively. According to NASA, they need to use π to 15 decimal places (3.141592653589793) for interplanetary navigation.

According to Wikipedia, the following is a timeline with regards to the accuracy of calculations of π:

top of page