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Approximate Value of Pi

Pi, $\pi $ is a mathematical term of uttermost importance in various studies such as statistics, probability, cosmology, mechanics, electromagnetism, and many more. Pi can be defined as the ratio of the circumference and the diameter of a circle, and is approximated to $\frac{22}{7}$ or 3.14159. Till date, the exact value of pie has not been obtained. Pi is an irrational number which is non-repeating and non-ending. For astronomical calculations, the approximation is done up to 39 places to get the accurate results.

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Archimedes's approximation of pi

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Archimedes's found the area of a circle by finding the area of two polygons, the polygon in which the circle is inscribed and the polygon which is inscribed in the circle. These areas are computed using the Pythagoras theorem. Archimedes was the first person to approximate the value of pi and calculating the areas gives the upper and lower bound of the area of the circle.

Archimedes's statement on the measurement of circle says that the ratio of circumference to the diameter of the circle will always be greater than $3\frac{1}{7}$ and less than $3\frac{10}{71}$.

Find approximation of value of pi

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1. Let us take a unit circle.

2. Draw a line with unit intervals.

3. Start rolling the unit circle on the line by putting the radius on 0.

4. The point at which the circle will complete a single round is somewhere near 3.14.

Approximate Value of Pi  

Practical approximations

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The most commonly used approximation of pi is $\frac{22}{7}$. There is one more approximation of pi which is $\frac{355}{113}$ which is correct up to six places. There are various algorithms being evolved to find the exact value of pi out of whom none has got complete success yet.
For calculations we need to use approximate value of pi. Some of the approximations we commonly use for pi are:

1) Integral approximation which is equal to $3$ (which is less commonly used in calculations).

2) As fractional approximation, we make use of fraction $\frac{22}{7}$ for pi.

3) As decimal we can use n digits after decimal being endless representation. So till $10$ places after decimal the value of pi is $3.1415926535...$

These are the most commonly used approximations of pi in numerical. The value to be chosen completely depends either what is mentioned in the given problem or else as per requirement. But the priority to use which one of these depends on what is mentioned in the problem.

Development of efficient formulae

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Since years, many people have been trying to approximate the value of pi. Many efficient formulae have been developed for this.

1. Polygon approximation of a circle: It was the approximation done by Archimedes.

2. The Machine like formulae: It was the approximation done by Daniel shank by using two different power series.

3. Classical formulae: Many mathematicians like Liu hui, Euler and Ramanujam worked on approximating the value of pi.

4. Modern algorithms: Algorithms like Borwein's and Gauss-Legendre have been used in by modern researchers for getting the value
of pi.
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