## What is the Area of a Circle?

circle is a simple closed shape in Euclidean geometry.

It is the set of all points in a plane that is at a given distance from a given point, the center. It is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the center is called the radius.

A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.— Euclid. Elements Book

The ratio of a circle’s circumference to its diameter is π (pi), is approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:

We were from the school days, that the area of a circle is found by the formula

Have you ever given a chance  to your students or yourself to know how this formula has been derived?

I think most of the readers here are aware of the area of a parallelogram. If yes, then you can proceed further (else go back and read about the area of a parallelogram and come back).

Here, we go

Let’s do a small activity to find the area of the circle.

First, begin by drawing a conveniently sized circle on a piece of cardboard. Now, divide the circle into 16 equal arcs. This may be done by marking off consecutive arcs of 22.5° or alternatively by consecutively dividing the circle into two parts, then four parts, then bisecting each of these quarter arcs, and so on.

It would look like the image shown below:

Fig. Circle divided into 16 equal parts

At last, the 16 parts or sectors, shown above, are then cut apart and placed in the manner
shown in the figure below.

Fig. Sectors (16 parts ) are arranged in the form of a parallelogram

From the above figure, we have base length of a parallelogram as $\displaystyle Base=\frac{C}{2}$

Where, $\displaystyle C=2\pi r$  r: radius of the circle.

We know that the area of the parallelogram is equal to the product of its base and altitude (which here is r).

i.e $\displaystyle =\left( \frac{C}{2} \right).r$

$\displaystyle =\left( \frac{2\pi r}{2} \right).r$

$\displaystyle =\pi {{r}^{2}}$

Hurray!! You derived the formula to find the area of a circle.

Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic” – Robert Chambers