IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: x. Volume 10, Issue 4 Ver. I (Jul-Aug. 2014), PP

Transcription

1 IOSR Journl of Mthemtics (IOSR-JM) e-issn: , p-issn: x. Volume 10, Issue Ver. I (Jul-Aug. 01), PP -8 Pi tretment for the constituent rectngles of the superscribed squre in the study of exct re of the inscribed circle nd its vlue of Pi (SV University Method * ) R.D. Srv Jgnndh Reddy Abstrct: Pi vlue equl to is derived from the Exhustion method of Archimedes (0 BC) of Syrcuse, Greece. It is the only one geometricl method vilble even now. The second method to compute is the infinite series. These re vilble in lrger numbers. The infinite series which re of different nture re so complex, they cn be understood nd used to obtin trillion of decimls to with the use of super computers only. One unfortunte thing bout this vlue is, it is still n pproximte vlue. In the present study, the exct vlue is obtined. It is A different pproch is followed here by the blessings of the God. The res of constituent rectngles of the superscribed squre, re estimted both rithmeticlly, nd in terms of of the inscribed circle. And vlue thus derived from this study of correct reltionship mong superscribed squre, inscribed circle nd constituent rectngles of the squre, is exct. Keywords: Circle, digonl, dimeter, re, rdius, side, squre I. Introduction Squre is n lgebric geometricl entity. It hs four sides nd two digonls which re stright lines. A circle cn be inscribed in the squre. The side of the squre nd the dimeter of the inscribed circle re sme. This similrity between dimeter nd side, hs mde possible to find out the exct length of the circumference nd the exct extent of the re of the circle, when this interreltionship between circle nd its superscribed squre, re understood in their right perspective. The difficulty is, the inscribed circle is curvture, though, its dimeter/ rdius is stright line s in the cse of side, digonl of the squre. When we sy different pproch is dopted, it mens, these re entirely new to the literture of mthemtics. The universl cceptnce to the new principles observed in the following method is tough job nd tkes time. However, s the following resoning wys re cent percent in ccordnce with the known principles, understnding of the ide is esy. To study the different dimensions, such s, circumference nd re of circle, constnt is inevitble. Similrly, to understnd perimeter nd re of the squre, nd re dopted nd hence, no constnt similr to circle is necessry in squre. In the present study, the re of the squre is divided into five rectngles. The res of rectngles re clculted in two wys: they re: 1. Arithmeticl wy nd. In terms of of the inscribed circle. Finlly, the rithmeticl vlues re equted to formuls hving, nd the vlue of is derived ultimtely, which is exct. II. Procedure Drw squre nd its two digonls. Inscribe circle in the squre. 1. Squre ABCD, AB Side. Digonls AC BD 3. O Centre, EF dimeter side. The circumference of the circle intersects two digonls of four points: E, H, F nd G. Drw prllel line IJ to the sides DC, pssing through G nd F. 5. OG OF rdius / 6. Tringle GOF. GF hypotenuse OG GF * This uthor studied B.Sc., (Zoology s Mjor) nd M.Sc., (Zoology) during the yers in the Sri Venkteswr University College, Tirupti, Chittoor district, Andhr Prdesh, Indi. And hence this uthor s mrk of his grtitude to the Alm Mter, this method is nmed fter University s Honour. Pge

2 Pi tretment for the constituent rectngles of the superscribed squre in the study of exct re of.. 7. IJ side 8. DI IG FJ JC 1 9. JC JB CB CJ Side hypotenuse JC, CB side 10. Bisect JB twice of CB side of Fig- JB JL + LB JK + KL + LM + MB 8 16 IJ GF 11. Similrly, bisect IA twice, of AD side of Fig- IA IP + PA IQ + QP + PN + NA 1. Join QK, PL, nd NM. 13. Finlly, the ABCD squre is divided into five rectngles. DIJC, IQKJ, QPLK, PNML nd NABM Out of the five rectngles, the uppermost rectngle DIJC is of different dimension from the other four bottomed rectngles. 1. Are of DIJC rectngle DI x IJ 15. The lower four rectngles re of sme re. For exmple one rectngle IQKJ IQ x QK 16. Are of rectngles Pge

3 Pi tretment for the constituent rectngles of the superscribed squre in the study of exct re of.. IQKJ + QPLK + PNML + NABM 17. Are of the squre ABCD DIJC + bottomed rectngles Prt-II 18. Let us repet tht Are of the ABCD squre d Are of the inscribed circle ; where dimeter side 19. When side dimeter 1 Are of the ABCD squre 1 x 1 1 d 11 Are of the inscribed circle 0. Corner re in the squre (of Figs 1,, nd 3) Squre re circle re 1 1. It is true tht ny bottomed rectngles, is equl to the corner re of the squre of Figs 1, nd 3. Thus, bottomed rectngle corner re Let us prove it i.e. S. No. 1 Prt-III 3. The inscribed circle is equl to the sum of the res of upper most rectngle DIJC S.No. 1 nd next lower 3 rectngles IQJK, QPLK nd PNML, nd ech is equl to Are of the inscribed circle Are of the corner region (S.No. 0) Are of the inscribed circle + corner re squre re + 5. The sum of the res of bottomed rectngles Squre re Uppermost rectngle DIJC where of S.No. 15 of 6 Pge

4 Pi tretment for the constituent rectngles of the superscribed squre in the study of exct re of.. nd S. No. 1 this is equl to S.No As the re of the corner region is equl to ny one of the bottomed rectngles, then it is (S.No. 0 & 1) 7. Then the sum of the res of bottomed rectngles 8. Finlly, Are of the uppermost rectngle DIJC Squre re bottomed rectngles 9. CJ length 3 3 Side AB IJ 30. Are of the upper most rectngle DIJC 3 3 CJ x IJ 31. Thus, the res of five rectngles which re interpreted in terms of bove, re Uppermost rectngle DIJC 3 bottomed rectngles Are of the ABCD squre Uppermost rectngle + bottomed rectngles 3 Are of the inscribed circle Uppermost rectngle DIJC + 3 bottomed rectngles 3 3 This is the end of the process of proof. 3. As the corner re is equl to 1. Arithmeticlly 16 nd. in terms of then S.No. 0 S.No. 1 where 1 III. Conclusion It is well known, tht is the formul to find out re of squre or rectngle. In this pper besides, formule, in terms of, of the inscribed circle in squre, re obtined, nd equted to the clssicl rithmeticl vlues of. One hs to dmire the Nture, tht, circle s re cn lso be represented exctly equl, by the res of rectngles, thus, the rithmeticl vlues of these rectngles, re equted to tht of circle, 1 which thus give rise to new vlue This uthor stnds nd bow down nd 7 Pge

5 Pi tretment for the constituent rectngles of the superscribed squre in the study of exct re of.. dedictes this work to the Nture. The Nture is the visible speck of the infinite Cosmos. The Cretor exists in the invisible Energy form of this infinite Cosmos. We cll this Cretor s GOD nd this uthor offers himself, surrenders himself totlly nd prys to THE LORD of the Cosmos of His/ Hers/ It s infinite goodness, s n infinitesimlly, smll living moving body, s mrk of humble grtitude to THE LORD. References [1]. Lennrt Berggren, Jonthn Borwein, Peter Borwein (1997), Pi: A source Book, nd edition, Springer-Verlg Ney York Berlin Heidelberg SPIN []. Alfred S. Posmentier & Ingmr Lehmnn (00),, A Biogrphy of the World s Most Mysterious Number, Pge. 5 prometheus Books, New York [3]. RD Srv Jgnnd Reddy (01), New Method of Computing Pi vlue (Siv Method). IOSR Journl of Mthemtics, e-issn: , p-issn: Volume 10, Issue 1 Ver. IV. (Feb. 01), PP 8-9. []. RD Srv Jgnnd Reddy (01), Jesus Method to Compute the Circumference of A Circle nd Exct Pi Vlue. IOSR Journl of Mthemtics, e-issn: , p-issn: Volume 10, Issue 1 Ver. I. (Jn. 01), PP [5]. RD Srv Jgnnd Reddy (01), Supporting Evidences To the Exct Pi Vlue from the Works Of Hippocrtes Of Chios, Alfred S. Posmentier And Ingmr Lehmnn. IOSR Journl of Mthemtics, e-issn: , p-issn: Volume 10, Issue Ver. II (Mr-Apr. 01), PP 09-1 [6]. RD Srv Jgnnd Reddy (01), New Pi Vlue: Its Derivtion nd Demrction of n Are of Circle Equl to Pi/ in A Squre. Interntionl Journl of Mthemtics nd Sttistics Invention, E-ISSN: P-ISSN: Volume Issue 5, My. 01, PP [7]. RD Srv Jgnnd Reddy (01), Pythgoren wy of Proof for the segmentl res of one squre with tht of rectngles of djoining squre. IOSR Journl of Mthemtics, e-issn: , p-issn: Volume 10, Issue 3 Ver. III (My-Jun. 01), PP [8]. RD Srv Jgnnd Reddy (01), Hippocrten Squring Of Lunes, Semicircle nd Circle. IOSR Journl of Mthemtics, e- ISSN: , p-issn: Volume 10, Issue 3 Ver. II (My-Jun. 01), PP 39-6 [9]. RD Srv Jgnnd Reddy (01), Durg Method of Squring A Circle. IOSR Journl of Mthemtics, e-issn: , p- ISSN: Volume 10, Issue 1 Ver. IV. (Feb. 01), PP 1-15 [10]. RD Srv Jgnnd Reddy (01), The unsuitbility of the ppliction of Pythgoren Theorem of Exhustion Method, in finding the ctul length of the circumference of the circle nd Pi. Interntionl Journl of Engineering Inventions. e-issn: , p- ISSN: , Volume 3, Issue 11 (June 01) PP: [11]. RD Srv Jgnndh Reddy (01), Pi of the Circle, t Pge