Pyramidization of Polygonal Prisms and Frustums

Transcription

1 European International Journal of cience and ecnology IN: Pyramidization of Polygonal Prisms and Frustums Javad Hamadani Zade Department of Matematics Dalton tate College Dalton, Georgia Introduction I ave taugt a geometry course for teacers at a Cristian college and a tate college in te past few years. opics covered in tis course include measurement, transformation geometry, plane and space figures, prolem solving in geometry, metods and materials for teacing geometry [7]. e relationsip etween te formulas for volumes of solids as always een difficult for some students wo took te aove course. For example, it was difficult for tem to see tat te volume of a pyramid is one tird of te volume of a prism wit equal ase and eigt. ey ad to eiter memorize tis relationsip or e ale to see tis relationsip y actually visualizing tree pyramids fitting in te prism of te same ase and eigt. e visualization was difficult for tem, and I designed a series of metods for dividing prisms into pyramids [9] to elp tem see te relationsip etween teir volumes. en we extended our division metods to truncated square pyramids. e Geometer s ketcpad was used in tese exercises. In tis paper I descrie te various sectioning metods we used to see te relationsip etween te volumes of polygonal pyramids wit volumes of polygonal prisms and truncated pyramids. e students realized tat prisms and truncated pyramids can e divided into polygonal pyramids; te same way tat polygons can e divided into triangles. Pyramids are made of triangular lateral faces and a polygonal ase tat can e divided into triangles. erefore, pyramids play te same role in tree-dimensional geometry tat triangles do in te plane. ese metods present practical exercises for etter understanding polygonal space figures. pecifically, students divided some polygonal prisms and a truncated square pyramid into polygonal pyramids and proved tat te volumes of te resulting pyramids add up to te volume of te original solids, a tedious ut wortwile exercise. owing tat te sum of te volumes of te resulting pyramids equals te volume of te original solid requires algeraic and sometimes trigonometric computations. e students were faced wit prolem-solving situations tat tey ad created y dividing prisms and truncated pyramids into pyramids and were callenged to find teir volumes and realize tat te sectioning metods were valid procedures. I elieve tat space figures, volumes and surface areas are difficult topics for teacers to teac. is may e due to te fact tat students ave difficulty wit te properties of triangle and oter geometrical figures in te plane and wen it comes to tree dimensions tey find tem more difficult. Wen prospective teacers wo take a course in geometry actually divide a prism or truncated pyramid into pyramids using te Geometer s ketcpad and wen tey calculate volumes of te resulting pyramids and see tat teir sum 88

2 European International Journal of cience and ecnology ol. No. April, 04 equals te volume of te original solids teir visualization of tree dimensions would e increased. ey can sare tese exercises wit teir students and teac tem visualization. e content of tis paper can e useful in elping teir students to overcome some of teir difficulties as far as te tree dimensional ojects are concerned. Proofs in Geometry Most proofs in geometry require us to sow tat line segments are congruent or angles are congruent. In order to do tis we must identify two congruent triangles and sow tat te line segments or te angles are te corresponding parts of tese congruent triangles. o identify te congruent triangles, occasionally we must draw auxiliary segments to form triangles. All tis process as to e done visually as we set pencil on paper and make drawings. e visualization or imagination of te situation sould precede any veralization and drawing for te proof. As in any learning activity, visualization as to e taugt and geometry and tree dimensional figures are ric areas for practicing visualization. e simple metods of dividing prisms into pyramids given in tis paper provide students wit prolem-solving situations and practicing tree-dimensional visualization. Polygons into riangles We can divide any polygon into triangles in many ways. For example: () Join any vertex of a polygon to oter vertices. Figure (a) () Join any interior point of a polygon to te vertices. Figure () In te first metod te numer of triangles will e n wit n > sides. In te second metod te numer of triangles will e n, te numer of sides of te polygon. (a) () Figure Cues into Pyramids Let us generalize te aove metods of dividing polygons to a cue of ase lengt. y te first metod, te cue can e divided into tree identical olique pyramids of ase lengt y joining a vertex to all te oter vertices. ee Figure (a). 89

3 European International Journal of cience and ecnology IN: / (a) () Figure e volume of te cue wit side is. erefore, te volume of eac square pyramid in Figure (a) is. In Figure (), for simplicity, we ave joined te center of te cue to eac vertex and eac pyramid formed on eac face as volume. quare Prisms into Pyramids e relationsip etween te volume of a square prism (cuoid, or rectangular parallelepiped) and square pyramid of te same eigt is demonstrated y filling a plastic model of te pyramid wit water or sand and emptying it into te plastic model of te prism, [Posamentier/mit/tepelman 00, p. 9]. e student will see tat eac time one-tird of te prism is filled, and to fill te prism we must repeat tis process tree times. erefore, pyramid =, () were, prism =, and te factor for te volume of pyramid will e rememered. ee Figure. 90

4 European International Journal of cience and ecnology ol. No. April, 04 (a) Figure () Metod A way to divide te square rigt prism is similar to te way tat we divided te cue; y joining a vertex to all oter vertices. ee Figure 4. Figure 4 In tis case, te sum of te volumes of te tree pyramids is: + + = + + =. 9

5 European International Journal of cience and ecnology IN: Remark We notice tat as in te case of te cue, a pyramid is formed on te lower ase, and te upper ase is divided into two congruent triangles and eac of tese triangles is a face of one of te oter two pyramids. Metod Anoter way to divide te rigt square prism into olique pyramids so tat two of te pyramids ave square ases is sown in Figure, elow. (a) () (c) (d) Figure In Figure, te two triangular pyramids of vertical eigts, Figure (c), are formed inside te prism wen we cut te two square pyramids, Figure () (wit ases and eigts and eac of volume ). We can put te two triangular pyramids togeter to form a rectangular pyramid, Figure (d) as sown and its volume is. o see te two triangular pyramids tat are formed inside te prism, paper models of te pyramids were made and put togeter as in Poto. One student remarked tat we could make a model of te square prism wit play-do, ten cut te square pyramids and see te two triangular pyramids wic will e left. 9

6 European International Journal of cience and ecnology ol. No. April, 04 Metod We can join an interior point of te square rigt prism to all its vertices and divide it into six pyramids, one on eac face. ee Figure. is is a generalization of te polygonal division into triangles, Figure (). O / Figure In Figure, we ave joined te midpoint of te altitude to all vertices to simplify te computation of te volumes of te six pyramids tat are formed. One rectangular pyramid is formed on eac face; and tere are four of tem, and eac as volume =. e oter two pyramids ave square ases and eac as a volume of. eir sum will e ( ) =, te volume of te square prism, wit ase side and vertical eigt. Pentagonal Rigt Prism Next, let us consider a regular pentagonal prism of ase side lengt s and eigt wose volume is: = as, () Were, is te eigt of te prism, s is te ase side lengt, and a is te apotem of te pentagonal ase in Figure 7. In dividing te polygonal prisms into pyramids regularity is not necessary, and we are assuming tat te ase is a regular pentagon to simplify computation of te area of te ase and ence te volume formula () Metod We can divide a pentagonal prism into polygonal pyramids y joining any point of te altitude, for example, te mid point of te altitude to te vertices of te lower ase and te vertices of te upper ase, as in Figure 7. is is again a generalization of dividing a polygon into triangles Figure (). We descrie tis metod first ecause trigonometry can e avoided in computations of te volumes. 9

7 European International Journal of cience and ecnology IN: a / a r s Figure 7 Consequently, te aove pentagonal prism is divided into two pentagonal pyramids wit eigts, and five rectangular pyramids wit ase lengts s and and eigts te apotem a of te pentagon; a total of seven pyramids, altogeter. In a polygonal prism wit n ase sides, tis metod results in n + pyramids e two pentagonal pyramids and te five rectangular pyramids are sown in Figure 7, aove. We will sow tat te volumes of tese seven pyramids add up to te volume of te prism, formula (). 94 Let us denote te area of eac pentagonal ase y A, te area of eac lateral face y A, te volume of eac of te pentagonal pyramids wit eigt, y, and te volume of eac of te rectangular pyramids wit ase area A, ase sides s,, and eigt a, y. We see tat A = ( as) and A = s and a + = A + A a = ( as) + ( s ) = as + as = as = e aove procedure can e applied to any polygonal prism.

8 European International Journal of cience and ecnology ol. No. April, 04 Metod We can also divide te pentagonal prism y joining a vertex of one of te ases to all oter vertices, as seen in Figure 8, elow. We note tat tis is a generalization of dividing a polygon into triangles y joining a vertex to all oter vertices Figure (a). ere are four pyramids in figure 8. One is pentagonal wit ase ACDE, te lower ase of te prism and vertical eigt, te eigt of te prism, and te oter tree are rectangular pyramids wit ases, CC, A A and AEE A all wit ase sides s and. Eac of te rectangular pyramids as a face tat is one of te triangles tat te upper pentagonal ase of te prism is divided into, a generalization of te remark made in te case of te square prism. It remains to sow tat te volumes of tese four pyramids add up to te volume of te pentagonal prism. First, we express a and s in terms of r. Were, r is te radius of te circumscried circle of te pentagonal ase. ee Figure 7. a = r cos s = r sin o o () It is necessary to use trigonometry since te rectangular pyramids ave vertical eigts tat depend on te sine of te exterior angle of te triangular face. In te following, to simplify computations, esides formula () we also use te doule-angle formula for sine, o o o sin 7 = sin cos (4) E' D' A' C' ' E A D C s Figure 8 9

9 European International Journal of cience and ecnology IN: e pyramid wit ase ACDE as volume = as = as. e pyramid wit ase CC o o as altitude s sin7 = r sin sin7 o, were we ave sustituted for s using formula (). erefore, its volume is o o = r sin sin 7 s o o o = r sin sin 7 r sin 4 o o = r sin sin 7 We ave sustituted r sin o for s in te second equation aove and simplified te result. e pyramid wit ase A A as altitude r + a = r + r cos o o = r( + cos ) We ave used formula () again to sustitute for a. Its volume is r s o = ( + cos ) o o = r( + cos ) r sin o o = r ( + cos ) sin Finally, te volume of te pyramid wit ase AEE' A' is te same as te one wit ase CC, wic is. Applying te doule-angle identity (4), te alf-angle identity, sin o o cos7 =, o o o o and te identity for sine of supplementary angles, sin44 = sin( ) = sin, we proceed to simplify te sum of tese volumes as follows: + + = r sin7 o + 4 o r sin sin7 o + o r ( + cos ) sin o 9

10 European International Journal of cience and ecnology ol. No. April, 04 8 = r sin 7 + r sin sin7 + r sin + r cos sin o 4 o o o o = r sin 7 + r ( cos 7 ) sin 7 + r sin + r sin 7 7 o 4 o 4 o o o = r sin 7 + r sin 7 r cos7 sin7 + r sin o o o = r sin 7 r sin44 + r sin o o o o = r sin7 r sin + r sin = r sin7 o o o o = r ( sin cos ) = ( r cos )( r sin ) = as. e last expression is te volume of te pentagonal prism, formula (). o o o o o o runcated quare Pyramid We now consider a truncated square pyramid (frustum) of lower ase lengt, upper ase lengt and vertical eigt. ee Figure 9, elow. We note tat a truncated square pyramid is not a prism, since te parallel ases are not congruent. However, our division procedures demonstrated aove can e applied in tis case ecause te volume formula for te truncated pyramid is te sum of volumes of pyramids. e formula is: = ( + + ). () I ave given a proof of formula () wic appeared in [8], for oter proofs and generalizations, see [], [], [], and [] in te References. Z W X Figure 9 97

11 European International Journal of cience and ecnology IN: Metod As in te cases of a cue, square, and pentagonal prisms, we can divide te truncated square pyramid y joining a vertex of a ase to all oter vertices as sown in Figure 0, elow. is again is a generalization of dividing a polygon into triangles y joining a vertex to all oter vertices, Figure (a). Z X X W Z W Figure 0 W We find te volume of eac olique pyramid in figure 0 to see if tey add up to te volume formula (), aove. e volume of te pyramid wit te red trapezoidal ase WX is ( + ). e volume of te one wit te lue square ase Z is. Finally, te volume of te lank ase pyramid wit ase te trapezoid W is also ( + ). Let us add tese tree volumes: ( + ) + + ( + ) = = + + = ( + + ). e last expression is, te volume of te truncated square pyramid, formula (). 98

12 European International Journal of cience and ecnology ol. No. April, 04 Metod We can also divide te truncated square pyramid in te following way, Figure elow, ecause te volume formula () is te sum of volumes of two square pyramids and one rectangular pyramid. Z W X (a) () (c) X W (d) Figure As in te case of te square rigt prism, once we cut te two olique square pyramids wit ase, te lower ase, and ase, te upper ase of te truncated pyramid, Figures (a) and (c), two triangular pyramids Figure (d) wit te ase triangle WX, and te ase triangle X and altitudes and, respectively, are formed inside te truncated pyramid. e volume of te first triangular pyramid is =, and te volume of te second is =. e sum of tese two volumes is, + =, wic is te middle term in formula () for te volume of te truncated square pyramid. Metod In Metod we apply a generalization of polygonal division Figure (); te procedure outlined and demonstrated for prisms. We divide te truncated square pyramid into six pyramids y joining te midpoint of te altitude to all vertices. 99

13 European International Journal of cience and ecnology IN: In Figure elow, te midpoint of te altitude, O, is joined to te vertices of te upper ase and te vertices of te lower ase, and six pyramids are formed. We sow tat tese six pyramids ave volumes tat add up to formula (). Z O W X Figure e lower and upper pyramids wit ases and, respectively, and eigts ave volumes = and =. e four pyramids tat are formed on te faces eac ave altitude 4 ( + ). o te volume of eac is = ( + ) ( + ) = ( + ) 4 4 = ( + + ) 4 erefore, te sum of tese four volumes is 4 = ( + + ). Let us add all te six volumes = + + ( + + ) = + + = ( + + ). e last expression is formula (), te volume of te truncated square pyramid. riangular Prisms Finally, we consider a triangular prism. Its division into and pyramids is given elow. Metod e parallel ases of a triangular prism are triangles and we can only join te vertex P of te lower ase to two vertices of te upper ase, and. en we join vertex of te upper ase to vertex R of te 00

14 European International Journal of cience and ecnology ol. No. April, 04 lower ase. ree triangular pyramids are formed wic ave equal volumes (immons 987). ee Figure, elow. R P R P R P P Q Q Figure Metod Metod is a generalization of dividing polygons into triangles y joining an interior point of te triangular prism to all te vertices. e triangular prism is divided into pyramids; two triangular and tree rectangular, one on eac face. ee Figure 4. We can sow tat teir volumes add up to te volume of te triangular prism te same way as in te square or pentagonal prisms. O R P Q Figure 4 Conclusion tudents evaluation of te aove division metods was positive. ey all said tat teir knowledge of tree-dimensional geometry was increased and tey learned ow to find volumes of pyramids tat make up a prism or a truncated pyramid. Moreover, tey learned tat we can divide polygonal prisms and truncated pyramids into polygonal pyramids in two general ways. First, we can join any interior point to te vertices of eac face. In tis way, a rectangular pyramid is formed on eac face inside te prism. ere are as many rectangular pyramids as tere are faces. wo polygonal pyramids, one on eac ase, are also formed. imilarly, all truncated polygonal pyramids can e divided into pyramids, and te pyramids formed 0

15 European International Journal of cience and ecnology IN: on te faces are trapezoidal. In ot cases, te numer of pyramids will e n +, were n is te numer of sides of te ase. econd, if n >, we can join a vertex of a polygonal prism or a vertex of a truncated polygonal pyramid to all oter vertices and tis metod will result in n polygonal pyramids, were n is te numer of sides of te ase. One of te ases will e divided into n triangles. Eac of tese triangles will e te face of a rectangular pyramid formed on a face in te case of a prism, or a trapezoidal pyramid formed on a face in te case of a truncated pyramid. ere will e a polygonal pyramid wit ase te ase of te prism or te ase of te truncated polygonal pyramid, and te total is ( n ) + = n.. In te special case wen te prism is triangular, n =, it is possile to divide it into pyramids as in Figure. References. Clay, Roert E. olume of te Frustum Generalized Matematics eacer 04, no. (00): 0.. Howard, Cristoper A. Matematics Prolems From Ancient Egyptian Papyri. Matematics eacer 0, no. (009): Posamentier, Alfred.; mit, everly.; tepelman, Jay eacing econdary Matematics: ecniques and Enricment nits.7 t Edition, Pearson, 00: immons, G. F. Precalculus Matematics in a Nutsell: Geometry, Algera, rigonometry. Janson Pulications, 987;.. nyder, Mark Frustum of a Pyramid Revisited Matematics eacer 0, no. (009):7 8.. u, Wenjiang A volume Generalization Matematics eacer 0, no. 7 (00): ussy, A.., Gustafson, R. D. asic Geometry for College tudents. econd Edition. rooks/cole, 00; Zade, Javad Hamadani Egyptian Geometry Matematics eacer 0, no. (008): Zade, Javad Hamadani Pyramidization European International Journal of cience and ecnology. ol. No., Feruary 0, pp. 4. 0