INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE ABSTRACT:- Vignesh Palani University of Minnesota - Twin cities e-mail address - palan019@umn.edu 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 twodimensional 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 = 2πr then π should be greater than 3 which it is. Similarly, the area of the six sided polygon = 6 h = 6 = 2.598 The area of the circle is a bit more than area of the six sided polygon because the shortest distance between two points is always a straight line. Hence the area should be more than 2.598, so if area = π then π should be greater than 2.598 which it is. This was the basic proof given by Archimedes. After this there was many improvements in the proof of circumference and area of circle and finally a proof involving Calculus was formulated. The circumference of the circle was evaluated using integral calculus. For a circle, x(ɵ) = cosɵ, y(ɵ) = sinɵ = Ɵ + Ɵ Ɵ = sinɵ +cosɵ Ɵ = Ɵ = 2π The area of the circle was found out by using integration. Area of the circle = 2πr.dr = π As there is π in the formula, the formula for circumference and area of circle is not an accurate one but an approximate one. π is the ratio of circumference to diameter of a circle whose value is 22/7. Below is a different approach to prove the existing formula for both circumference and area of circle using more of geometry. In this proof a very small arc (which can be assumed to be a
straight line) of a circle of large radius was considered and a similar arc was drawn to the original arc to get a closed figure. The perimeter of the closed figure was found out using trigonometry. Then a condition (which makes the closed figure a circle) was substituted in the formula for perimeter of the closed figure to get the formula for circumference of the circle. Similarly, the area of the closed figure was found out using integration. Then a condition (which makes the closed figure a circle) was substituted in the formula for area of the closed figure to get the formula for area of the circle. PROOF FOR THE EXISTING FORMULA OF CIRCUMFERENCE OF CIRCLE:- Highly magnified figure of the arc AB C O O PROOF:- ABCDE is a circle. Angle AOB = θ. For very small angle θ, arc AB can be considered to be a straight line. By considering the highly magnified image of the arc, the length of the arc AB using trigonometry = 2Rsin$/2 For very small angle Ɵ, sinɵ is approximately Ɵ.
*+,( lim ( =lim (-./ / 1.2 ( ( 2 -.3 3. ( = lim ( 1 (7 +(8 9 (: ;.. On applying the limit, we get lim ( *+,( ( =1 Therefore, as θ approaches 0, sin$ $. Therefore length of the arc AB = Rθ. On applying Pythagoras theorem in triangle AOC, we get: = > +? == = +> 2=>+? == R=? +> /2> tan ( = B C-D = BD (On substituting the value of R) B 7 -D7 - BD θ = 2 tan B 7 -D 7 Perimeter of the shape AXBYA = 2Rθ = 4 B7 1D 7 D - BD tan B 7 -D 7 AXBYA will become a circle if a=b=r. On applying this condition the circumference of closed figure AXBYA = 4 7 1 7 = 2πr tan - 7 7-7 Hence the existing formula for circumference of a circle is derived. PROOF FOR THE EXISTING FORMULA OF AREA OF THE CIRCLE:-
C (0,0) PROOF:- The equation of the circle considering 'C' as the origin is =6> = Therefore, the area of the region under the arc AXB is B = 6 6=6>. -B =? = 6? = sin -B C 6=6>? The area of the closed figure AXBYA is twice the area of the region under the arc AXB. Area of the closed figure AXBYA = 2(? = 6? = sin -B C 6=6>?) = 2?FG B7 1D 7 D H 6? G B7 1D 7 D H sin - G BD -D 7 J B 7 1D 7H6BIB7 D as R = B7 1D 7 AXBYA will become a circle if a=b=r. On substituting this condition in the above formula, we get area of the closed figure AXBYA is π. Hence the existing formula for area of a circle is derived. DISCUSSION:- Many different methods for proving the existing formula of circumference and area of a circle have been found out in the past years. But the author's approach are entirely new and novel. The D
author's proof uses more of geometry rather than calculus in deriving the existing formula of circumference and area of the circle. REFERENCES:- K.Raghul Kumar, New proofs for the perimeter and area of a circle, The general science journal, Jan 10, 2010.