The number $\pi$ is the ratio of the circumference of a circle ($C$) to its diameter ($d$).

$\pi$ = $\frac{C}{d}$

It has been known since ancient times that this ratio is the same for all the circles. Using the value of $\pi$ one can simply determine the answer for complex problems in algebra. In this section we will be learning more about $\pi$

The first systematic efforts to find a numerical approximation for $\pi$ was made by Archimedes (240 B.C.), who proved that $\frac{22}{7}$ < $\pi$ < $\frac{223}{71}$ by finding the perimeter of the polygons inscribed in and circumscribed about a circle.

What is Pi

In about A.D. 480 the Chinese physicist Tsu Ch'ung-chih gave the approximation, Which is correct to six decimals.

$\pi$ = $\frac{355}{113}$ = $3.141592....$
 
This remained the most accurate estimation of $\pi$ until the Dutch mathematician Adrianus Romanus (1593) used polygons with more than a billion sides to compute $\pi$ correct to 15 decimals. The Englishman William Shanks spent 15 years (1858-1873) using infinite serious and trigonometric identical methods to compute $\pi$ corrected to 707 decimals. And later it was found that this figures were wrong beginning with the 528th decimal. Today with help of computers one can calculate $\pi$ value corrected to millions of decimal.       

The value of pi find more application in algebra. The following are examples which make use pi value for the calculation.
1. Formula for the area (A) of a circle also uses $\pi$

A = $\pi r^{2}$

2. Formula for the volume of sphere

Volume (V) = $\frac{4}{3}$ $\pi r^{3}$

3. Perimeter of a circle = $2 \pi r$


Solved Examples

Question 1: Find the area of a circle with radius 5 cm
Solution:
 
Given $r = 5 cm$ 

Area, $A$ = $\pi r^{2}$

= $3.14 \times (5cm)^{2}$

= $3.14 \times 25 cm^{2}$

= $78.5 cm^{2}$

 

Question 2: Find the volume of a ball whose radius is 3 inches.
Solution:
 
Given $r = 3$ inches 

Volume (V) = $\frac{4}{3}$ $\pi r^{3}$

                    = $\frac{4}{3}$ $\pi 3^{3}$ 

                   = $\frac{4}{3}$ $3.14 \times 27$

                               = $113.04$ in3