How to Calculate Pi 𝞹 Values? Calculating Pi 𝞹 Values Using Polygon Trigonometry | Science Experiment | Mathematics
Introduction: 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. 367
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. 367