Solids. Solids. Curriculum Ready.

Transcription

1 Curriulum Rey

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3 This ooklet is ll out ientifying, rwing n mesuring solis n prisms. SOM CUES The Som Cue ws invente y Dnish sientist who went y the nme of Piet Hein. It is simple 3 # 3 # 3 ue tht is split up into seven unique piees. These n e put together in mny ifferent wys to form ue n other soli shpes. The seven piees re speil euse no other omintions of 3 or 4 ues joine y their sies re possile. Even relly ool furniture hs een me using these soli omintions. Look them up to fin out more out them. lso serh out Polyominoes for the flt shpe equivlent. Q Use ruler n penil to hnge the plne shpe elow into soli, 3-imensionl rwing. Work through the ook for gret wy to o this H 14 1 SERIES TOPIC

4 How oes it work? Polyherons Polyherons re lose 3-imensionl solis forme with flt polygon fes n stright eges. Prts of polyheron: Corner/Vertex (verties plurl) Sie/Fe Ege Visile Ege (soli line) Ege not visile from this view (otte line) Polyheron? Polyheron? Polyheron? - ll fes re flt polygons - Soli is lose - Not ll fes re flt polygons - Soli is lose - ll fes re flt polygons - Not lose soli Here re just few of the mny si types of polyherons tht exist. Prisms These solis hve the sme shpe (ross-setion) ll long their length. Nme y the ross-setion shpe. Pltoni polyherons ll fes re the sme regulr polygon. n equl numer of eges meet t eh vertex. Right-Prisms The en ross-setionl shpe is perpeniulr (90⁰) to the length of the soli. Pyrmis ll eges from the verties of the se meet t point. Nme y the se polygon. Convex polyherons stright line rwn through these will only pss through two fes. Conve polyherons stright line rwn through these n pss through more thn two fes. The ifferent views elow show how stright lines n pss through Convex or Conve polyherons. Convex Conve 1 2 View from the top stright line n only pss through two fes View from the top stright line rwn through the soli n pss through more thn two fes. 2 H 14 SERIES TOPIC

5 How oes it work? Polyherons s with polygons, polyheron n e nme using Greek prefixes mthing the numer of fes it hs. Polyheron nming n lssifition hrt Fes Nme Soli Pltoni (ll eges equl length) 4 Tetrheron Forme using 4 equilterl tringles 5 Pentheron N/ 6 Hexheron Cue: forme using 6 equl squres 7 Heptheron N/ 8 Otheron Forme using 8 equilterl tringles Some solis o not hve pltoni form in the tle. In ft, there re only 5 pltoni solis in totl! The other two re: Doeherons (forme using 12 regulr pentgons) Iosherons (forme using 20 equilterl tringles) There re only 5 euse the totl internl sum of the polygon ngles whih meet t eh vertex must e no more thn 360⁰. Tetrheron Hexheron Otheron Doeheron Iosheron 60 ⁰ 60⁰ 60 ⁰ 108 ⁰ 3 # 60⁰ < 360⁰ Euler s rule for onvex polyherons Euler isovere tht for ll onvex polyherons, the sum of the ll the fes n verties, minus the numer of eges will lwys equl 2. For this ox, the totl numer of: fes (F) = 6 F + V - E = 2 verties (V) = 8 eges (E ) = 12. Fes Verties 3 # 90⁰ < 360⁰ Eges 4 # 60⁰ < 360⁰ 3 # 108⁰ < 360⁰ ` F + V - E = = 2 5 # 60⁰ < 360⁰ H SERIES 14 TOPIC 3

6 POLYHEDRONS *POLYHEDRONS *POLYHEDRONS * How oes it work? Polyherons 1 Ientify whih of these solis in rel life woul e lssifie s polyherons or not. n of rink soli house rik 12-sie ie Polyheron Polyheron Polyheron Not polyheron Not polyheron Not polyheron n Ornge tresure hest n open ox Polyheron Polyheron Polyheron Not polyheron Not polyheron Not polyheron 2 Tik ll the est-mthing properties tht eh of these polyherons hve. Convex Conve Pltoni Prism Right-Prism Pyrmi Convex Conve Pltoni Prism Right-Prism Pyrmi Convex Conve Pltoni Prism Right-Prism Pyrmi Convex Conve Pltoni Prism Right-Prism Pyrmi e Convex Conve Pltoni Prism Right-Prism Pyrmi f Convex Conve Pltoni Prism Right-Prism Pyrmi 3 Show tht Euler s rule works for eh of the polyherons elow. Tetrheron Fes = Pentgonl pyrmi Fes = Verties = Verties = Eges = Eges = + - = + - = Pentgonl Prism Otheron Doeheron e.../.../ = + - = + - = 4 H 14 SERIES TOPIC

7 How oes it work? Volume of prisms using unit ues Polyherons ll tke up spe in the rel 3-imensionl worl. The mesurement of the spe they oupy is lle the Volume (or pity for liquis). unit ue is polyheron where every fe is squre with sies one unit of mesurement long. 1 unit Little shes on eh sie men they re ll the sme length. Volume (V) = 1 ui unit = 1 unit 3 (in shorter, units form) The volume of the prism elow is foun y ounting the numer of whole unit ues use to uil it. 1 unit Volume () = 4 ui units = 4 unit 3 Here re some prisms inluing hlves n qurters of unit ue: Clulte the volume (V) of these solis (i) Volume (V) = 4 whole ues + 3 hlves of ue = 4 ui units + 3 # 2 1 ui units = ( ) ui units 1 unit = 5.5 units 3 When units of mesurement re given, they re use inste of the wor units (ii) V = 6 whole ues + 4 hlves of ue + 6 qurters of ue = 6 ui mm + 4 # 2 1 ui mm + 6 # 1 4 ui mm = ( ) ui mm 1 mm = 9.5 mm 3 H 14 5 SERIES TOPIC

8 How oes it work? Volume of prisms using unit ues.../.../ VOLUMEOFPRISMSUSINGUNITCUES * 1 Clulte the volume of these prisms forme y omining 1 unit ues: V = whole ues V = whole ues 1 unit = units 3 1 m = m 3 V = mm 3 V = m 3 1 mm 1 m e f V = units 3 V = m unit Clulte the volume of these prisms forme y omining whole n hlf unit ues: 1 m 1 mm V = whole + hlves of ue = mm # 2 mm 3 = mm 3 1 m V = whole + hlves of ue = m # 2 m 3 = m 3 1 m V = m 3 V = m 3 1 m e f V = units 3 V = mm 3 1 unit 1 mm 6 H 14 SERIES TOPIC

9 How oes it work? Volume of prisms using unit ues 3 Fin the volume of these prisms forme y omining whole, hlf n qurter unit ues: 1 unit 1 m V = whole + qurters of ue V = whole + qurters of ue = m 3 + # 1 4 m3 = units 3 + # 1 4 units3 = m 3 = units 3 V = whole + hlves of ue + qurters of ue = m 3 + # 1 2 m3 + # 1 4 m3 1 m = m 3 V = whole + hlves of ue + qurters of ue = mm 3 + # 1 2 mm3 + # 1 4 mm3 1 mm = mm 3 4 Clulte the volume of these prisms: 1 unit 1 m 1 m 1 mm V = V = V = V = 5 Clulte the volume of this prism n ern yourself n wesome pssport stmp! 1 m H 14 7 SERIES TOPIC

10 Where oes it work? Volume of right-prisms Prisms hve the sme polygon ross-setionl shpe throughout their height/length. Eh of these slies re the ext sme shpe. height In right-prisms, the ross-setion polygon is perpeniulr to the height of the soli. Volume (units 3 ) = re of the ross-setion shpe (units 2 ) # height of the soli (units) V = # h The lultion for the volume is me esier euse the ross-setionl re is given For this right-prism, the ross-setionl re is 25 m 2 n the height of the soli is 4 m. (i) Volume (V) = re of the ross-setionl polygon # height of the soli = 25 m = 25 m 2 # 4 m 2 = 100 m 3 4 m This next one is not tully prism s it hs urve sies, ut we lulte its volume the sme wy. (ii) = 12 m 2 Volume (V) = re of the en shpe # height 5 m = 12 m 2 # 5 m = 60 m 3 ll mesurements must e in the sme units efore lulting the volume. (iii) Volume (V) = re of the ross-setionl polygon # height of the soli = 36 m 2 # 50 mm 36 m 2 = 36 m 2 # 5 m 50 mm = 180 m 3 8 H 14 SERIES TOPIC

11 Where oes it work? Volume of right-prisms 1 Complete the volume lultions for these right-prisms: = 12 m 2 2 m 8 m m 4.5 mm 2 14 mm V = m 2 # m Cross-setion Height = m 3 V = m 2 # m Cross-setion Height = m 3 V = mm 2 # mm Cross-setion Height = mm 3 2 Complete the volume lultions for these right-solis: 10 m 2 50 m 8.25 m 2 1 m 12.5 mm 2 40 m V = m 2 # m Cross-setion Height = m 3 V = m 2 # m Cross-setion Height = m 3 V = mm 2 # mm Cross-setion Height = mm 3 3 Use the re onversion tle to help omplete the volume lultions for these right-prisms: ' 100 ' squre millimetres mm 2 squre entimetres m 2 squre metres m 2 25 m m 2 5 mm 12 m V = mm 2 # mm V = m 2 # m # 100 # 1000 Cross-setion Height = mm 3 Cross-setion Height = m 3 4 Clulte the missing mesurements of these solis: = m 3 ' m h = m 3 ' m 2 12 m Volume Height 12.6 m 2 Volume re h Volume = 84 m = m 2 3 Volume = 14 m 3 = m = m 3 ' m h = mm 3 ' mm m 2 80 m = m 2 h = mm Volume = 14.8 m 3 Volume = 31 mm 3 H SERIES 14 TOPIC 9

12 *VOLUME OF RIGHT PRISMS*VOLUME OF RIGHT PRISMS Where oes it work? Volume of right-prisms 5 Clulte the volume for these right-prisms showing ll working. 16 m 2 28 mm 2 4 m 5 mm ` V = m 3 ` V = mm m m 2 7 m 8 m ` V = m 3 ` V = m 3 e f 15 m 2 80 mm ` V = m 3 4 m m ` V = m 3 g h m 2 2 m 2 20 m 10 mm ` V = mm 3 ` V = m 3 6 This pttern ws me using pver riks. Eh rik hs height of 105 mm n ross-setionl re of 288 m 2. Wht is the totl volume of riks in m 3 use to mke this 28-rik pttern (to 2 eiml ples)? psst: 1 m 3 = m 3.../.../ H 14 SERIES TOPIC

13 Where oes it work? Volume of right-prisms Volume lultions involving fluis nee nswers using units of pity Cpity onversion hrt ' 1000 ' 1000 ' 1000 ' 1 m 3 = 1 kilolitre (kl) 1000 m 3 = 1 Litre (L) 1 m 3 = 1 millilitre (ml) mirolitre μl millilitres ml litres L kilolitres kl meglitres ML # # 1000 # 1000 # 1000 (i) How mny litres (L) woul ontiner of this shpe hol? Volume (V) = 15 m 2 # 0.85 m = 15 m 2 85 m = m 3 `Cpity = kl = L 1 m 3 = 1 kilolitre (kl) 1 kl = 1000 L (ii) Clulte the ross-setionl re of this right-prism ontiner if it n hol 210 ml. Cpity = 210 ml = 210 m 3 1 m 3 = 1 millilitre (ml) m 2 re = Volume ' height of the soli 30 m = 210 m 3 ' 30 m = 7 m 2 7 Clulte the pity of these ontiners when full. 24 m 2 3 m Volume = m 3 25 m 30 m 2 Volume = m 3 ` Cpity = kl ` Cpity = ml 45 mm 884 mm 2 Volume = mm 3 ` Cpity = L Volume = m mm m ` Cpity = L 8 Clulte the ross-setionl re () of these right-prism ontiners if they oth hol 42.5 litres mm 85 m = m 2 = m 2 H SERIES 14 TOPIC 11

14 Where oes it work? Volume of Retngulr prisms For retngulr or squre right-prisms: Volume = Length # reth # Height V = l # # h units units units units 3 = 12 units 3 Joine together the shpe forme looks like this: height length reth Volume = Length # reth # Height 3 # 2 # 2 units 3 = 12 units 3 So you simply multiply ll the imensions of retngulr prism to lulte the volume. Clulte the volume of these retngulr prisms (i) (ii) 4 m 3 m Volume (V) = length # reth # height = 1.5 m # 3 m # 4 m = 18 m 3 efore you n use the volume formul, ll mesurements must e in the sme units. 50 mm 1.5 m 1.5 m V = length # reth # height = 50 mm # 1.5 m # 1.5 m = 5 m # 1.5 m # 1.5 m = m 3 Rememer: the orer you multiply the numers y oes not mtter. hnge to e ll sme units You oul hve lso hnge 1.5 m to 15 mm inste, then the volume woul look like mm 3 12 H 14 SERIES TOPIC

15 Where oes it work? Volume of Retngulr prisms 1 Clulte the volume of these right-prisms using the formul V = l # # h 3 m 3 m 4 m 6 mm 2.5 mm 5 mm V = m # m # m = m 3 V = mm # mm # mm = mm 3 5 m 7 m 220 mm 2 m 0.06 m 6 m V = m # m # m = m 3 V = m # m # m = m 3 e f 0.5 m 0.05 m V = mm # mm # mm = mm mm 150 mm V = m # m # m = m 3 2 Clulte the volume of these right-prisms showing ll your working. 3.5 m 6 mm 5 m 6 m 4.75 mm ` V = m 3 ` V = mm mm 40 mm 140 mm 0.18 m 2 m 8.5 m ` V = m 3 ` V = m 3 H SERIES TOPIC

16 Where oes it work? Volume of Retngulr prisms 3 Hollow retngulr onrete riks hve een use to uil the tringulr wll shown here. 25 m Empty Spe 32 m 22 m rik wll 24 m Eh rik Do you think there is more onrete or hollow spe in this wll? Mrk on the numer line where you think the lne of onrete to ir is in this wll. ll ir m equl ll onrete How mny onrete riks were use to uil this wll? Wht is the totl volume of this wll if eh rik ws not hollow? Clulte the volume of the spe tken up y the hole in eh rik? e Clulte the totl volume of hollow spe in this wll. f The totl volume of onrete = the volume of the wll - the volume of hollow spe. Clulte the volume of onrete in the wll. g How mny litres of onrete re require to mke ll the riks neee for this wll? h Using the informtion you now hve from your lultions, mrk new point on the numer line where the tul nswer for question is. H 14 SERIES TOPIC

17 Where oes it work? Volume of Retngulr prisms 4 The glss sreen on the front n k of one rn of smrt phone hs these imensions: 11.5 m 0.08 m 5.7 m Over five yer perio, this rn sol pproximtely 240 million ( ) phones. Mrk elow wht you preit the height of squre prism ontiner with with n length of 10 m woul nee to e to hol ll this glss. < 2.5 m 2.5 m or more ut 5 m or more ut 7.5 m or more ut > 10 m less thn 5 m less thn 7.5 m less thn 10 m Clulte the volume of glss in eh phone sreen in m 3. How mny ml of glss re there in eh phone sreen? How mny ml of glss re there in 240 million ( ) phones? e If ll this glss ws poure into prism ontiner with squre se of sie length 1 m (100 m), how high woul the ontiner nee to e to hol it ll?.../.../ VOLUMEOFRECTNGULRPRISMS*VOLUMEOFRECTNGULRPRISMS* H SERIES 14 TOPIC 15

18 Wht else n you o? Different views of solis It n e iffiult to show ll the fetures of soli shpe in single igrm. For exmple is the soli rwn here me using 5 or 6 ues? Viewe from the opposite iretion we see tht this oul represent ny one of these solis elow: 6 ues 6 ues 5 ues So to ommunite whih one it is extly n esily, we rw ifferent flt views to mke it ler. The three si views re, Sie n views. Sie With these views of the soli given, we n see lerly tht it is tully the first one me with 6 ues. Sie Opposite view nother wy to help imgine the pln views is y pretening there is glss ox roun the soli refleting the view of eh sie. 16 H 14 SERIES TOPIC

19 FDIFFERENT VIEWS OF SOLIDS*DIF Wht else n you o? Different views of solis 1 Mth the pln views elow with the orret soli to fin the nswer to this question:.../.../ ERENT VIEWS OF SOLIDS* Wht is the nme given to rwings tht show the front or sies of n ojet? Sie (v) C F M U (x) (ii) G T (vi) L O S (i) H W T (vii) E (viii) (iii) V Y P E (iv) (ix) N N K (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) H SERIES TOPIC

20 Wht else n you o? Different views of solis 2 Eh of these solis were forme using six ues only. Drw the front, top n sie views for eh one. Sie Sie Sie Sie Sie e Sie f Sie 18 H 14 SERIES TOPIC

21 Wht else n you o? Different views of solis 3 Drw the front, top n sie views for eh of 3-D imges me using smll prisms. Sie Sie Sie Sie Sie e Sie H SERIES TOPIC

22 Wht else n you o? Different views of solis The soli view n e rwn esily from the ifferent pln views using isometri gri pper The leing ege n e inite with letters to show the ege to e use t the front. Sie Sie 4 Use the isometri gris to rw the solis represente y the front, top n sie views given. Sie Sie Sie Sie 20 H 14 SERIES TOPIC

23 Wht else n you o? Different views of solis 5 Use the isometri gris to rw the letters represente y the front, top n sie views given. Sie Sie Sie Sie e Sie H SERIES 14 TOPIC 21

24 Wht else n you o? 3D rwings using vnishing points When ojets re fr wy they pper smller thn they tully re. We n use this ft to help trnsform flt 2-imensionl polygons into prisms. For exmple, let s trnsform this squre into prism. 1. Put ot some ple to the left/right n just ove the squre. This ot is lle the Vnishing Point (VP). VP 2. Rule some stright lines in penil from eh vertex on the sme sie s the VP, pssing through it. VP 3. Drw vertil n horizontl lines s shown elow to rete the fr sie of the soli. VP 4. Complete y rwing in the other eges long the vnishing point lines. VP 22 H 14 SERIES TOPIC

25 Wht else n you o? 3D rwings using vnishing points Use the given vnishing point for eh of these polygons to mke them prisms..../.../ VP *TRNSFORMINGFLTSHPESINTO3DUSINGVNISHINGPOINTS 2 VP 3 This one will nee two horizontl n two vertil lines in step 3. VP 4 This is triky one to finish off with. e reful with the horizontl n vertil lines step. VP H SERIES TOPIC

26 Chet Sheet Here is wht you nee to rememer from this topi on solis Polyherons Close 3-imensionl solis forme with flt polygon fes n stright eges. Corner/Vertex (verties plurl) Sie/Fe Ege Visile Ege (soli line) Ege not visile from this view (otte line) Prisms These solis hve the sme shpe (ross-setion) ll long their length. Nme y the ross-setion shpe. Pltoni polyherons ll fes re the sme regulr polygon. n equl numer of eges meet t eh vertex. Right-Prisms The en ross-setionl shpe is perpeniulr (90⁰) to the length of the soli. Pyrmis ll eges from the verties of the se meet point. Nme y the se polygon. Convex polyherons stright line rwn through these will only pss through two fes. Conve polyherons stright line rwn through these n pss through more thn two fes. Euler s rule for onvex polyherons F + V - E = 2 Fes Verties Eges Volume of right-prisms Volume of retngulr right- prisms V = # h height reth height length Volume = Length # reth # Height Different Views of Pln Views (or Elevtions) Sie 24 H 14 SERIES TOPIC

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