INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE
Vignesh Palani University of Minnesota - Twin cities e-mail address - [email protected]
ABSTRACT :-
In this brief work, the existing formulae of circumference and area of circle have been confirmed in a different approach using more of geometry.
KEYWORDS:-
Geometry, Circles, Circumference, Area, Calculus.
INTRODUCTION :-
Geometry is a very important branch in mathematics which stands for: 'geo' which means earth and 'metron' which means measurement. Through Geometry many important theorems in mathematics have been found out. Geometry is basically study of size, shape, orientation, of two- dimensional and three-dimensional figures. Geometry helps to develop skills in deductive thinking which is applied in all other fields of learning. Hence it plays an important role in both physics and chemistry. Artists use Geometry to create their masterpieces. We come across Geometry in our everyday life as we are surrounded by space and things of different shapes. A major contributor to the field of Geometry was Euclid-325 BC who is typically known as the Father of Geometry and is famous for his work "The Elements". The major achievement in Geometry was Euclidean geometry which was followed for several centuries. Then came the non-Euclidean Geometry which led to radical transformation of concept of space. Ever since the first invention on Earth, Geometry has always been a crucial step in the discovery of other important things on Earth. The distance around a circle is known as the circumference of the circle. The formula for circumference of circle was unknown for several years. Finally it was found out by Archimedes, a famous Greek Mathematician. There are many proofs for circumference of circle. The proof given by Archimedes several years ago is as follows:
He marked a point on the circumference of the black circle and considered it to be the centre of the red circle. The radius of both the circle is the same. Hence the edge of the red circle should touch the centre of the black circle. He drew the line segment connecting the centers of both circles which showed the radius of both the circles. He drew three diameters across the red circle, as shown in the figure, and he completed the polygon. This polygon consists of six equilateral triangles. The perimeter of the polygon is six times the radius of the circle (r) which is equal to 6r. The circumference of the circle is a bit more than perimeter of the polygon because the shortest distance between two points is always a straight line. Hence the circumference should be more than 6r, so if circumference = 2 πr then π should be greater than 3 which it is. Similarly, the area of the six sided polygon = 6 × ℎ