Question

Why do we use \[\pi \] for finding the area of a circle?

Answer 1

Hint: Pi (π) is the ratio of the circumference of a circle to its diameter. It doesn't matter how big or small the circle is - the ratio stays the same. Properties like this that stay the same when you change other attributes are called constants.

Complete step-by-step answer:

In Geometry, for computing the area, perimeter or any other property of a certain shape, we need to depend upon the dimensions of the shape. For eg. In case of a Square, we depend on its side length. In case of a Rectangle, we use its length and breadth.

Circle being a curved shape, it was difficult to depend upon a certain dimension. It was observed that the circles' area and circumference depends upon its diameter. Initially, the circles' circumference was measured roughly using threads. The ratio of the circumference and the diameter, i.e. (Circumference/Diameter) among the circles was observed to be similar but not equal. Aryabhatta defined the irregularity of this ratio to be an Irrational Number. He also found the value of this ratio upto 5 significant figures as 3.1416.

Hence \[\pi =\dfrac{circumference}{diameter}......(1)\]

So finding circumference from equation (1) and using the information that radius is half of diameter we get,

\[circumference=\pi \times 2\,radius......(2)\]

We also know that \[Area=\dfrac{circumference}{2}\times radius.....(3)\]. So now substituting the value of circumference from equation (2) in equation (3) we get,

\[Area=\pi \times {{(radius)}^{2}}\]


Note: We have to read the introduction part of the chapter circle to answer this question. We can measure \[\pi \] by constructing a physical wheel and rolling it out - but we won't get more than a digit or two of accuracy.