How to Calculate Pi 𝞹 Values? Calculating Pi 𝞹 Values Using Polygon Trigonometry | Science Experiment | Mathematics

As we all know that the circle is just a polygon, with infinite sides. So, with this concept, we can calculate the value of pi 𝞹. With simple trigonometry. We will consider every object as a circle & polygon at the same time. And use the naming convention interchangeably so, don't get confused.

Supplies

Trigonometry

Mathematics

Step 1: Circle Is a Polygon With Infinite Sides

We can see in the above figure. As we go on increasing the number of sides of the polygon its shape starts to look like a circle.

When the number of sides of the polygon is equal to infinity it becomes a perfect circle.

Step 2: Pi 𝞹 Vs. Sides

In the graph shown above. It is a graph of the 𝞹 pi values calculated versus the number of sides of the polygon used to calculate that value. The more the number of sides the more accurate the value we get approximated to 3.141592653589793238.

Step 3: Polygons With Diagonals

The above figure shows the diagonals of the polygon.

The number of sides of polygon = number of diagonals.

For a perfect circle, there will be infinite diagonals. And the length of the diagonal will be equal to the diameter of the circle.

So, we will consider the radius = digonal/2.

Step 4: Decagon for Calculations

Let us consider this decagon for the explanation and calculation of the 𝞹 (pi) value.

It has 10 sides & 10 diagonals.

Step 5: Angles (θ) of Polygon

Let us consider the angle between any two diagonals of the polygon as theta(θ).

Theta = 360 / no of sides

for decagon :

θ = 360/ 10

θ = 36°

Step 6: Consider This Triangle

Let us consider this triangle for the calculation.

Step 7: Triangles in Polygon

Every polygon has diagonals. So a triangle is formed between two diagonals and the side.

So, we can use this triangle-based method for any polygon to calculate the value of pi.

Step 8: Triangle for Calculations

This is the triangle to be used for calculation. We have separated it out from polygon. For better visualization and easier calculations.

Step 9: Specs of Triangle

There is an angle θ theta between two diagonals. The two sides of the triangle can be considered radii because it is equal to the diagonal/2 of the polygon. The other side of the triangle is equal to the edge of the polygon.

Step 10: Calculating Theta Θ

The values we know are theta, radius, and phi.

To find: base,side

let us consider raduis = 10

θ = 360 / no of sides

for decagon :

θ = 360 / 10

θ = 36°

Step 11: Calculating Phi Φ

As per the figure shown. The phi value is equal to half of the theta value.

Φ = θ /2

for decagon:

Φ = 36 /2

Φ = 18°

Step 12: Calculating Base of Triangle

This is a right-angled triangle.

Therefore base = radius x sin(Φ)

for decagon:

base = 10 x sin(18)

base = 3.09016994

Step 13: Calculating Edge of Polygon

The side length of the polygon is equal to twice the base of the right-angled triangle.

side length = 2 x base

for decagon:

side length = 3.09016994 x 2

side length = 6.18033988

Step 14: Calculating Perimeter of Polygon

The circle is a polygon with infinite edges.

The perimeter of any polygon = no of sides x length of the side

for decagon:

perimeter = 10 x 6.18033988

perimeter = 61.8033988

Step 15: Calculating 𝞹 Pi Value

We are considering a circle as a polygon with infinite sides.

So, the perimeter of the polygon is equal to the circumference of the circle.

circumference of circle = 2 x 𝞹 x radius

2 x 𝞹 x radius = no of sided x length of side

𝞹 = (no of side x length of side)/(2 x radius)

That's how we calculated the value of 𝞹.

for decagon:

𝞹 = (10 x length of side)/(2 x 10)

𝞹 = (10 x 6.18033988)/(2 x 10)

𝞹 = 3.09016994

Step 16: Increasing Sides Reduces Error

In the graph shown above. It is a graph of the 𝞹 pi values calculated versus the number of sides of the polygon used to calculate that value. The more the number of sides the more accurate the value we get approximated to 3.14159.

The table has the values of the number of sides of the polygon and the respective pi value calculated from that figure.