THE POSSIBILITY OF ESTIMATING THE VOLUME OF A SQUARE FRUSTRUM USING THE KNOWN VOLUME OF A CONICAL FRUSTRUM

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1 THE POSSIBILITY OF ESTIMATING THE VOLUME OF A SQUARE FRUSTRUM USING THE KNOWN VOLUME OF A CONICAL FRUSTRUM SAMUEL OLU OLAGUNJU Adeyemi College of Education NIGERIA lagsam04@aceondo.edu.ng ABSTRACT Tis paper considers te calculation of te volumes of a Conical Frustrum and a Square Frustrum, noting ow related tey are and ow te knowledge of one can elp in estimating te oter. An earlier paper ad noted from records tat te Egyptians used a process of dividing te pyramid into two portions first, calculating te areas as A and A, and ten obtaining te volume of te pyramid as one-tird te eigt multiplied by te sum of te two different areas A and A added to te square-root of te product of te two areas {i.e. V A A A A }. Tis paper reduces te cumbersome nature of suc calculations. It was noted tat Macrae et al (00) gave te volume of a Conical Frustrum, ere designated as V as V CF CF ( R Rr r ) ; F being te Volume of Square-based Frustrum; R, te Radius of te large Circular base; r, te Radius of te small Circular top; and = te eigt of te Conical Frustrum. In te process of developing a less-cumbersome model for te volume of a Squarebased frustrum, Olagunju (0) considered formulas for complete Pyramids, including tat of Circular Frustrum, to arrive at a proven formula for a Square Frustrum as V SF ( D Dd d ) Were V SF Volume of Square Frustrum, D = Diagonal of te large Square base, d = diagonal of te small Square top, and = te eigt of te Square Frustrum. Te consideration process confirmed te possibility of estimating te volume of one given te oter, if te Top and Base Diagonals of te Square Frustrum respectively ave equal lengts wit te Top and Base Diameters of te Conical Frustrum, Keywords: Volume, Pyramids, Frustrum, Conical, Square, Diameters, Diagonals.0 Introduction Te basis of progress in any endeavor in life, especially scientific progress, lies in improvement. Tis is wy everybody yarns for improvement. Similarly, te essence of education is to find a way of improving on an earlier situation. Tus, learning and mastering te use of existing models will not be sufficient, as it is more useful to see ow suc models could be improved upon for te benefit of mankind, and at times it could be useful to establis te relationsip between two models. Tis forms te basis on wic te establised formulae for te Volume of a Conical (or Circular-based) Frustum and tat of a Square (or Square-based) Frustum were considered for te purpose of establising a relationsip between te two, approacing te issue from teir respective Diameters and Diagonals..0 Purpose of te Study Te essence of tis work is to establis te possibility of estimating te volume of Square frustums from tat of a known Conical frustum (and vise-versa), provided tat certain given condition is fulfilled. Tis Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 7

2 will elp teacers and students of Matematics as well as constructing establisments to safe some time in teir effort in estimating suc..0 Some Necessary Clarifications Te following clarifications sould please be noted. Pyramids and Frustums According to Hart (005), as noted in Olagunju (0), a Pyramid is a Polyedron aving one polygonal face (called base ) and all oter faces as Triangles, meeting at te Vertex (called Apex ). A special kind of Pyramid wose base is circular and all slant-edge lines meet at te vertex is referred to as a Cone or a Conical Pyramid. Wen a part of a Pyramid is copped off troug te apex, it becomes a Truncated Pyramid, usually referred to as a Frustum.. Classification of Pyramids Pyramids are classified by teir dimensions. Wile a Regular pyramid is one wit a base wit regular polygon (e.g. Square-Based, Rectangular-Based), a Rigt pyramid is one wose apex is joined to te center of te base by a perpendicular line. Anoter type wit one single cross-sectional sape aving lengts scaling linearly wit its eigt is referred to as an Arbitrary pyramid.. Pyramidal Frustums A Truncated Pyramid is one wose part as been copped-off to a given eigt. Frustums are named after te sape of teir base. Wile a Square Frustum is one wose Base and Top are bot in te form of Squares (usually, te lengt of one end-face is smaller tan te oter), a Conical Frustum (Truncated Cone) is one wose Base and Top are circular (usually, te radius of one end-face is smaller tan te oter). 4.0 Related Existing Models 4. Volume of a Conical Frustum Considering an existing model of te Conical Frustum were te volume of te Frustum is obtained by subtracting te copped-off top volume from te big cone volume, we ave te Volume of a big cone as V BC R H ; and te Volume of te copped-off cone as V SC r x, Going by Weisstein s (00) Concise Encyclopedia of Matematics, irrespective of te base sape or position of te apex relative to te base, Pyramidal volume is V Ab were Ab = Area of te base, and = te eigt (perpendicular distance of te apex from te base) remembering tat te capacity of te pyramid equals one-tird of a cylinder of same eigt and same base-radius. Ten, according to Kalejaiye et al (00), te volume of te Conical Frustum V CF so formed is given by V CF R H r x were R = radius of Big Cone, H = Heigt of Big Cone, r = radius of Small Cone, and x = eigt of Small Cone. Tis leads to Hero s formula, V CF ( R Rr r ). Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 7

3 4. Volume of Pyramids As earlier discussed, and as observed by Harris and Stocker (998), te Volume of a pyramid is given as one-tird of te product of base-area and perpendicular eigt. i.e. Volume = (base-area x eigt) Tus, considering a Square Pyramidal Frustum, if l = b (square base.) and base area = l x b, Ten, VP l x b x l b From te above, we obtain te volume of a Truncated Square Pyramid tus: If te volume of a Big Square-based Pyramid is V BP L H V SP l And if te volume of te copped-off small Pyramid is x Ten, te volume of te Truncated Square-based Pyramid (i.e. Pyramidal Frustum) so formed is given by V PF = V BP - V SP V PF L H l x ( L H l ) Were: l = lengt of Small-Square-Top (Base of copped-off top pyramid), x = eigt of Small Pyramid L = Lengt of Big-Square-Base Pyramid, H = Heigt of Big Pyramid [(H = x + ), = eigt of Frustum],. 4. Some Guiding Principles for te Square-Based Model Axiom I: Since te original Pyramid as a Square base, ten, te top copped-off Small Pyramid also as a Square base. Axiom II: Te ratio of te eigt of te top copped-off pyramid to te eigt of te original big pyramid equals te ratio of te diagonal of te top copped-off pyramid to te Diagonal of te original big pyramid. x : H = d : D Lemma I: Since te Big Base-Square Lengt and te Small Base-Square lengt are in te ratio L : l, and tis affects teir diagonals, ten, D : d = L : l 4.4 Volume of a Square Frustum Let te base and top diagonals of te pyramidal frustum be D and d respectively. If its eigt is, and te Volume is designated as V SF, Volume of Square Frustum will be te difference between te Volume of large Square Pyramid and Volume of copped Square Pyramid. Considering Diagram.4 below, were L = Lengt of Big-Square-Base, l = lengt of Small-Square-Base, x = eigt of Small Pyramid, = eigt of Frustum, H = Heigt of Large Pyramid, (eigt of small pyramid + eigt of Frustum) Wic implies tat te Heigt of Pyramid PEFGI = + x = H Area of Square base EFGI = L. L = L Heigt of Pyramid PJKMN = x Area of Square base JKMN = l. l = l Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 74

4 4.4: Diagram I: Pyramid P x N d H J M K F D E G I If te volume of a Large Square Pyramid is And if te volume of te copped smaller Pyramid is V BP L H V SP l Ten, volume of te Square Frustum formed is F VBP P ( L H l ) Now, Volume of Frustum JKMNEFGI is given by Volume of Pyramid PEFGI Volume of Pyramid PJKMN = Volume of PEFGI Volume of PJKMN i.e. VJKMNEFGI = VPEFGI - VPJKMN Tus, VJKMNEFGI = L ( x ) l x (4.) If D is te diagonal of te larger square EFGI above, And d is te diagonal of te smaller square JKMN, Ten, by Pytagoras, From PJM, d l l d (4.) And From PEG, D L L D (4.) Tus, substituting (4.) and (4.) in (4.), we ave Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 75

5 D d VJKMNEFGI = x x D x d x D D x d x = = = D x( D d ) (4.4) x x But considering similar triangles PJM and PEG, d D xd xd d or x( D d) d d Hence, x (4.5) D d Substituting (4.5) in (4.4), we ave d VJKMNEFGI = D ( D d)( D d) D d = D d( D d) = D Dd d = D Dd d (4.) Equation (4.) is te lagsamolu formula obtained for te Square Frustum. Tis was well illustrated and found useful and less-cumbersome. 5.0 Establising a Relationsip between Square Frustum and Conical Frustum 5. Additional Guiding Principles for te Relationsip Axiom III: Te respective Top and Base Diameters of te Conic Frustum must be equal to te respective Top and Base Diagonals of te Square Frustum. Axiom IV: From axiom III, it follows tat te ratio of te respective Top Diameter to Base Diameter of te Conic Frustum must be same as te ratio of te Top Diagonal to te Base Diagonal of te Square Frustum. Lemma II: Since te Big Radius Lengt and te Small Radius lengt are in te ratio R : r, tis affects teir diameters. i.e., ten, R : r =D : d 5. Relationsip between te Volumes of te Two Frustums: Considering diagrams (5.) and (5.) were te Top and Base Diagonals of te Square Frustum in (5.) are equal to te Top and Base Diameters of te Conical Frustum in (5.), we consider teir Volumes as follows: From diagram (5.), we note tat Volume of Square Frustum = D Dd d (5.) From diagram (5.), we also note tat Volume of Conical Frustum = ( ) R Rr r (5.) Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 7

6 5.: Diagrams II Consider te diagrams below: 5. Square Frustum: d D 5. Conical Frustum: d D Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 77

7 But R D Hence, equation (5.) becomes D D d d Conical Frustum Volume = D Dd d = = 4 D Dd d = D Dd d (5.) By Axiom III, Top Diagonal of Square Frustum = Top Diameter of Conical Frustum Base Diagonal of Square Frustum = Base Diameter of Conical Frustum Tus, Conical Frustum Volume = D Dd d = D Dd d = Volume of Square Frustrum (5.4) Hence, it is noted from equation (5.4) tat te Volume of a Conical Frustum equals alf te product of Pi ( ) and te Volume of a Square Frustum..0 Illustrations Here, we attempt to illustrate tis finding by considering certain situations below:. Illustration I Consider te situation were r:r = d:d = : If = 5, r 8, R ( r), Rr R r, Ten, = 5, d ( r), D ( R) 8, Dd D d 4 Ten, we ave it tat ( R Rr r ) = 5 ( 8) = 80 (.) D Dd d 5 0 = (8 4 ) = 0 Multiplying by, we ave Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 78

8 0 80 (.) Hence, from (.) and (.), it is clear tat te Volume of a Conical Frustum equals alf te product of Pi ( ) and te Volume of a Square Frustum.. Illustration II Consider te situation were r:r = d:d = : If = 5, r 8, R ( r) 7, Rr R r 4, Ten, = 5, d ( r), D ( R) 88, Dd D d 9 Ten, we ave it tat ( R Rr r ) = 5 (7 4 8) = 50 (.) D Dd d = 5 (88 9 ) = 080 Multiplying 080 by, we ave (.) Hence, from (.) and (.), it is clear tat te Volume of a Conical Frustum equals alf te product of Pi ( ) and te Volume of a Square Frustum.. Illustration III Consider te situation were r:r = d:d = :4 If = 5, r 8, R (4 r) 8, Rr R r, Ten, = 5, d ( r), D ( R) 5, Dd D d 8 Ten, we ave it tat ( R Rr r ) = 5 (8 8) = 840 (.) D Dd d = 5 (5 8 ) = 0 Multiplying 0 by, we ave Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 79

9 0 840 (.) Hence, from (.) and (.), it is clear tat te Volume of a Conical Frustum equals alf te product of Pi ( ) and te Volume of a Square Frustum..4 Illustration IV Consider te situation wen = 0.8, d 0.8, D 0.7, Dd 0. Ten, we ave it tat = 0.85, r 0.045, R 0.8, Rr 0.09 ( R Rr r ) = 0.8 ( ) = (.4) D Dd d = 0.8 ( ) = 0.8 Multiplying 0.8 by, we ave (.4) Hence, from (.4) and (.4), it is clear tat te Volume of a Conical Frustum equals alf te product of Pi ( ) and te Volume of a Square Frustum..5 Illustration V Consider te situation involving measurements less tan. If =.7, d 0.7, D.5, Dd.88 Ten, we ave it tat =.7, r 0.8, R.88, Rr 0.7 ( R Rr r ) =.7 ( ) =.40 (.5) D Dd d =.7 ( ) =.804 Multiplying.804 by, we ave (.5) Hence, from (.5) and (.5), it is clear tat te Volume of a Conical Frustum equals alf te product of Pi ( ) and te Volume of a Square Frustum. Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 80

10 5. Illustration VI Consider te situation wit decimals greater tan. wen = 0., d 8.0, D.5, Dd 45.0 Ten, we ave it tat = 0., r 4.5, R 8.5, Rr.5 ( R Rr r ) = 0. ( ) = (.) D Dd d = 0. ( ) = 98.5 Multiplying 98.5 by, we ave (.) Hence, from (.) and (.), it is clear tat te Volume of a Conical Frustum equals alf te product of Pi ( ) and te Volume of a Square Frustum. Conclusion: Given te above analysis and illustrations, tis now confirms te existence of a strong relationsip between te Volume of a Conical Frustum and tat of a Square Frustum, indicating tat alf te product of Pi ( ) and Volume of a square Frustum equals te Volume of a Conical Frustum, and tat te known volume of one can terefore be used to estimate te unknown volume of te oter, provided tat te Top and Base radii of te Conical Frustum are of te same ratio as te respective Top and Base diagonals of te Square Frustum. Precaution: Since every model as its own precaution(s), it sould be noted tat tis finding can only be used successfully if it is ascertained tat te Top and Base radii/diameters of te Conical Frustum are respectively of equal ratio as te Top and Base diagonals of te Square Frustum. Oterwise, it may fail. Recommendation: It is recommended tat te students and establisments willing to estimate te volume Square frustum may now do so conveniently troug te volume of a Conical frustum wic tey already know, and vise-versa provided tat te said ratios are as given above. References Harris, J. W. and Stocker, H. (998): "Pyramid." Handbook of Matematics and Computational Science. (P ). Springer-Verlag, New York. Hart, G. (005): Pyramids, Dipyramids, and Trapezoedra. ttp:// com/virtualpolyedra/pyramids-info.tml. Kern, W. F. & Bland, J. R. (948): Pyramid and Regular Pyramid: Solid Mensuration wit Proofs (nd ed.). New York, NY: Wiley & Sons. Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 8

11 Macrae, M. F.; Kalejaye, A. O.; Cima, Z. I.; Garba,G. U.; Ademosu, M.; Canon, J. B.; Smit, A. M.; Head, H. C. (00): New General Matematics for Senior Secondary Scools Bk I ( rd Edition). England: Pearson Education Limited. Olagunju, S. O. (0) Volume of a Square-Based Frustum: Alternative Formula (lagsamolu Equation). In Nwakpa, Izuagie and Akinbile (Eds) Meeting te Callenges in Science Education. (P.8 9). Babson Press, Ondo. Weisstein (00): CRC Concise Encyclopedia of Matematics. (P. 55, 5, 404). Proceedings of INCEDI 0 Conference ISBN: t-st August 0, Accra, Gana 8

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