New proofs for the perimeter and area of a circle

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Beverley Priscilla McCormick
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1 New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com Abstact: In this bief wok, the authos confimed the existing fomulae fo the peimete and aea of a cicle in a diffeent appoach. Key Wods: Classical geomety, Staight Lines, Tiangles, Cicles, Peimete, Aea. MSC: 51 M04 PACS: 0.40.D Intoduction: Geomety stands fo: geo which means eath and metia which means measue (Geek). A majo contibuto to the field of geomety was Euclid 35 BC who is typically known as the Fathe of Geomety and is famous fo his woks called The Elements. As one pogesses though the gades, Euclidian geomety (Plane Geomety) is a big pat of what is studied. Howeve, non-euclidean geomety will become a focus in the late gades and college math. Simply put, geomety is the study of the size, shape and position of dimensional shapes and 3 dimensional figues. Howeve, geomety is used daily by almost eveyone. In geomety, one exploes spatial sense and geometic easoning. Geomety is found eveywhee: in at, achitectue, engineeing, obotics, land suveys, astonomy, sculptues, space, natue, spots, machines, cas and much moe. When taking geomety, spatial easoning and poblem solving skills will be developed. Geomety is linked to many othe topics in math, specifically measuement and is used daily by achitects, enginees, achitects, physicists and land suveyos just to name a few. In the ealy yeas of geomety the focus tends to be on shapes and solids then moves to popeties and elationships of shapes and solids and as 1

2 abstact thinking pogesses, geomety becomes much moe about analysis and easoning. Geomety is in evey pat of the cuiculum K -1 and though to college and univesity. Since most educational juisdictions use a spialing cuiculum, the concepts ae e-visited thoughout the gades advancing in level of difficulty. Typically in the ealy yeas, leanes identify shapes and solids, use poblem solving skills, deductive easoning, undestand tansfomations, symmety and use spatial easoning. Thoughout high school thee is a focus on analyzing popeties of two and thee dimensional shapes, easoning about geometic elationships and using the coodinate system. Studying geomety povides many foundational skills and helps to build the thinking skills of logic, deductive easoning, analytical easoning and poblem solving to name just a few. Some of the tools often used in geomety include: Compass, potactos squaes, gaphing calculatos, geomete s sketchpad, ules etc. In this aticle we ae going to see a poof that aea and peimete of a cicle ae not accuate but only appoximate. π / π O 0 ө B 3 4 A Conside a cicle with adius and cente at O. In the cicle, conside the cuve path AB

3 which is vey vey small in length. Any vey small potion of a cuved path can be appoximated as a staight line Using the above concept, the cuved path AB is appoximated as a staight line. As AB is a vey shot line, angle AOB is vey vey small and then OAB is appoximated as a ight angled tiangle, ight angle at A and B. i.e. Angle OAB = Angle OBA = 90 o (appoximately) (1) Let d = Angle AOB = Angle BOA () The ight angled tiangle is shown below. O dө B In the above figue A Angle OAB = Angle OBA = 90 o (appoximately) and d = Angle AOB = Angle BOA Also AB AB Sind OA OB (3) Let AB = dp and OA = OB =. Then dp Sind (4) As d <<< 1 d = 0 o (appoximately) (5) As d = 0 o (appoximately) Sind d [1-3] (6) Fom (4) and (6), dp d ( appoximately) i.e. dp d (appoximately) (7) 3

4 (7) is the equation fo the length of the cuved path AB. Integating equation (7) as vaies fom 0 to, we get the peimete of the cicle p d (appoximately) (8) 0 i.e. p (appoximately) (9) Aea of the Cicle Conside the appoximations Angle OAB = Angle OBA = 90 o (appoximately) and Angle AOB = Angle BOA is a vey small angle. Aea of the ight angled tiangle OAB = 1 x base x height = 1 x AB x OA = 1 x dp x (10) Use (7) in (10), we get Aea of ight angled tiangle OAB, da = 1 d (appoximately) (11) Integating equation (11) as vaies fom 0 to, we get Aea of the cicle, 1 A = d (appoximately) 0 i.e. Aea of the cicle, A = (appoximately) (1) 4

5 Since the ight angles Angle OAB and Angle OBA and the angle at the cente, Angle AOB ae appoximate, the length of the ac AB i.e. dp and the aea of the tiangle OAB i.e. da ae also appoximate not accuate. Hence the peimete and the aea of a cicle ae not accuate but only appoximate values. Discussion: Thee ae diffeent ways fo both peimete and aea of cicle. But the autho s findings ae entiely new and novel. The simila appoach may be attempted in othe aeas of geomety and mathematics. The existing fomulae wee geometically deduced. But the autho s esults wee deived geometically and tigonometically using calculus. Refeences: [1] chapte 10, pp. 19 [] Richad Couant (1956), Diffeential and Integal Calculus, Blackie & Son, London, Vol.1, pp [3] Ewin Keyszig (1979), Advanced Engineeing Mathematics, John Wiley, New Yok, pp

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