Nyob hauv tsev kawm ntawv ntawm cov khoom geometry, ib qho ntawm cov duab yooj yim tshaj plaws uas tsis muaj qhov loj me ntawm peb lub spatial axes yog lub quadrangular prism. Xav txog hauv tsab xov xwm seb nws yog dab tsi, cov ntsiab lus nws muaj, thiab koj tuaj yeem suav nws thaj chaw thiab ntim li cas.
Lub tswvyim ntawm lub prism
Hauv geometry, prism yog ib daim duab spatial, uas yog tsim los ntawm ob lub hauv paus zoo ib yam thiab sab saum toj uas txuas rau sab ntawm cov hauv paus no. Nco ntsoov tias ob lub hauv paus tau hloov pauv mus rau ib leeg siv kev ua haujlwm ntawm kev txhais lus sib luag los ntawm qee qhov vector. Txoj hauj lwm ntawm lub prism no ua rau lub fact tias tag nrho nws sab yog ib txwm parallelograms.
Tus naj npawb ntawm sab ntawm lub hauv paus tuaj yeem ua tau raws li qhov xav tau, pib los ntawm peb. Thaum tus lej no zoo li tsis muaj qhov kawg, lub prism smoothly tig mus rau hauv lub tog raj kheej, txij li nws lub hauv paus dhau los ua lub voj voog, thiab sab parallelograms, sib txuas, tsim ib lub cylindrical nto.
Zoo li txhua yam polyhedron, lub prism yog tus cwj pwm los ntawmsab (dav hlau uas khi daim duab), ntug (ntu raws li ob sab sib tshuam) thiab vertices (lub rooj sib tham ntawm peb sab, rau ib tug prism ob ntawm lawv yog lateral, thiab lub thib peb yog lub hauv paus). Qhov ntau ntawm cov npe peb lub ntsiab ntawm daim duab yog sib cuam tshuam los ntawm cov lus hauv qab no:
P=C + B - 2
Ntawm no P, C thiab B yog tus lej ntawm cov npoo, sab thiab vertices, feem. Qhov kev qhia no yog lej lej ntawm Euler lub theorem.
Daim duab saum toj no qhia txog ob lub ntsiab lus. Ntawm lub hauv paus ntawm ib tug ntawm lawv (A) yog ib tug tsis tu ncua hexagon, thiab sab sab yog perpendicular mus rau lub hauv paus. Daim duab B qhia lwm qhov prism. Nws ob sab tsis yog perpendicular rau hauv paus, thiab lub hauv paus yog ib txwm pentagon.
Pris quadrangular yog dab tsi?
Raws li qhov tseeb ntawm cov lus piav qhia saum toj no, hom prism feem ntau txiav txim siab los ntawm hom polygon uas tsim lub hauv paus (ob lub hauv paus yog tib yam, yog li peb tuaj yeem tham txog ib qho ntawm lawv). Yog hais tias tus polygon no yog ib tug parallelogram, ces peb tau ib tug plaub fab prism. Yog li, txhua sab ntawm hom prism no yog parallelograms. Lub quadrangular prism muaj nws lub npe - parallelepiped.
Tus naj npawb ntawm sab ntawm ib qho parallelepiped yog rau, thiab txhua sab muaj qhov zoo sib xws rau nws. Txij li thaum lub hauv paus ntawm lub thawv yog ob sab, qhov seem plaub yog lateral.
Tus naj npawb ntawm vertices ntawm parallelepiped yog yim, uas yooj yim pom yog tias peb nco ntsoov tias cov vertices ntawm lub prism yog tsim tsuas yog nyob rau hauv lub vertices ntawm lub hauv paus polygons (4x2=8). Kev siv Euler's theorem, peb tau txais tus lej ntawm cov npoo:
P=C + B - 2=6 + 8 - 2=12
Tawm ntawm 12 tav, tsuas yog 4 tus tsim ntawm nws tus kheej los ntawm ob sab. Qhov seem 8 nyob rau hauv lub dav hlau ntawm lub hauv paus ntawm daim duab.
Ntxiv rau hauv kab lus peb yuav tham tsuas yog hais txog quadrangular prisms.
Type of parallelepipeds
thawj hom kev faib tawm yog cov yam ntxwv ntawm lub hauv paus parallelogram. Nws yuav zoo li no:
- tsis tu ncua, nws cov kaum tsis sib npaug 90 o;
- rectangle;
- ib square yog ib lub quadrilateral.
Qhov thib ob ntawm kev faib tawm yog lub kaum sab xis ntawm sab hla lub hauv paus. Muaj ob qhov xwm txheej sib txawv ntawm no:
- lub kaum sab xis no tsis ncaj, ces lub prism hu ua oblique lossis oblique;
- lub kaum sab xis yog 90o, ces qhov prism yog duab plaub los yog ncaj.
Qhov thib peb ntawm kev faib tawm yog cuam tshuam nrog qhov siab ntawm prism. Yog hais tias lub prism yog duab plaub, thiab lub hauv paus yog ib lub square lossis ib lub duab plaub, ces nws yog hu ua cuboid. Yog hais tias muaj ib lub xwmfab ntawm lub hauv paus, lub prism yog lub duab plaub, thiab nws qhov siab yog sib npaug rau qhov ntev ntawm sab ntawm lub square, ces peb tau txais lub cim lub voos xwmfab zoo.
Prism nto thiab thaj chaw
Cov txheej txheem ntawm tag nrho cov ntsiab lus uas nyob ntawm ob lub hauv paus ntawm prism(parallelograms) thiab ntawm nws sab (plaub parallelograms) ua rau saum npoo ntawm daim duab. Lub cheeb tsam ntawm qhov chaw no tuaj yeem suav nrog kev suav cov cheeb tsam ntawm lub hauv paus thiab qhov nqi no rau sab saum toj. Tom qab ntawd lawv cov lej yuav muab tus nqi xav tau. Ua lej, qhov no yog sau raws li hauv qab no:
S=2So+ Sb
Ntawm So thiab Sb yog thaj tsam ntawm lub hauv paus thiab sab saum toj, raws li. Tus naj npawb 2 ua ntej So tshwm vim muaj ob lub hauv paus.
Nco ntsoov tias cov qauv sau tau siv tau rau txhua qhov prism, thiab tsis yog rau thaj tsam ntawm plaub fab prism.
Nws yog ib qho tseem ceeb kom nco qab tias thaj tsam ntawm ib qho parallelogram Sp yog xam los ntawm tus qauv:
Sp=ah
Qhov twg cov cim a thiab h qhia qhov ntev ntawm ib sab thiab qhov siab kos rau sab no, raws li.
cheeb tsam ntawm ib daim duab plaub nrog lub hauv paus square
Nyob rau hauv ib lub quadrangular prism, lub hauv paus yog square. Rau qhov tseeb, peb qhia nws sab ntawm tsab ntawv a. Txhawm rau xam cov cheeb tsam ntawm lub quadrangular prism tsis tu ncua, koj yuav tsum paub nws qhov siab. Raws li cov lus txhais rau qhov ntau, nws yog sib npaug rau qhov ntev ntawm lub perpendicular poob los ntawm ib lub hauv paus mus rau lwm qhov, uas yog, sib npaug ntawm qhov deb ntawm lawv. Cia peb txhais nws los ntawm tsab ntawv h. Txij li thaum tag nrho cov ntsej muag yog perpendicular mus rau lub hauv paus rau hom prism nyob rau hauv kev saib xyuas, qhov siab ntawm ib tug quadrangular prism yuav sib npaug rau qhov ntev ntawm nws sab ntug.
BCov qauv dav dav rau thaj tsam ntawm ib qho prism yog ob nqe lus. Thaj tsam ntawm lub hauv paus hauv qhov no yooj yim los xam, nws yog sib npaug rau:
So=a2
Xaiv thaj tsam ntawm qhov chaw sab nraud, peb sib cav raws li hauv qab no: qhov saum npoo no yog tsim los ntawm 4 daim duab plaub zoo tib yam. Ntxiv mus, ob sab ntawm lawv txhua tus yog sib npaug rau a thiab h. Qhov no txhais tau tias thaj tsam ntawm Sb yuav muab sib npaug rau:
Sb=4ah
Nco ntsoov tias cov khoom 4a yog ib puag ncig ntawm lub hauv paus square. Yog tias peb nthuav dav qhov kev qhia no rau qhov xwm txheej ntawm qhov kev txiav txim siab, ces rau lub kaum sab xis ntawm lub qhov rooj sab nrauv tuaj yeem suav tau raws li hauv qab no:
Sb=Poh
Po yog ib puag ncig ntawm lub hauv paus.
Rov qab mus rau qhov teeb meem ntawm kev suav thaj tsam ntawm lub quadrangular prism tsis tu ncua, peb tuaj yeem sau cov qauv kawg:
S=2So+ Sb=2a2+ 4 ah=2a(a+2h)
Thaj chaw ntawm oblique parallelepiped
Xaiv nws yog qhov nyuaj me ntsis dua li lub duab plaub. Nyob rau hauv cov ntaub ntawv no, lub hauv paus cheeb tsam ntawm ib tug quadrangular prism yog xam siv tib lub mis raws li ib tug parallelogram. Cov kev hloov pauv cuam tshuam txog kev txiav txim siab ntawm thaj chaw sab nraud.
Yuav ua li cas, siv tib lub mis los ntawm ib puag ncig raws li tau hais hauv kab lus saum toj no. Tsuas yog tam sim no nws yuav muaj qhov sib txawv me ntsis. Tus qauv dav dav rau Sb nyob rau hauv rooj plaub ntawm oblique prism yog:
Sb=Psrc
Ntawm no c yog qhov ntev ntawm sab ntug ntawm daim duab. Tus nqi Psr yog qhov ib puag ncig ntawm daim duab plaub. Qhov chaw no yog tsim raws li hauv qab no: nws yog tsim nyog los hla tag nrho cov sab ntsej muag nrog ib tug dav hlau kom nws yog perpendicular rau tag nrho cov ntawm lawv. Cov duab plaub uas tau tshwm sim yuav yog qhov xav tau txiav.
Daim duab saum toj no qhia txog qhov piv txwv ntawm lub thawv oblique. Nws hla-hatched seem tsim cov kaum sab xis nrog rau sab. Qhov ib puag ncig ntawm ntu yog Psr. Nws yog tsim los ntawm plaub qhov siab ntawm lateral parallelograms. Rau qhov quadrangular prism no, thaj tsam sab nraud yog suav nrog cov qauv saum toj no.
Qhov ntev ntawm kab pheeb ces kaum ntawm lub cuboid
kab pheeb ces kaum ntawm parallelepiped yog ib ntu uas txuas ob lub vertices uas tsis muaj ob sab uas ua rau lawv. Tsuas muaj plaub kab pheeb ces kaum nyob rau hauv ib lub quadrangular prism. Rau ib lub cuboid nrog ib daim duab plaub ntawm nws lub hauv paus, qhov ntev ntawm tag nrho cov kab pheeb ces kaum sib npaug rau ib leeg.
Daim duab hauv qab no qhia txog daim duab sib xws. Cov ntu liab yog nws kab pheeb ces kaum.
Xaiv nws qhov ntev yog qhov yooj yim heev, yog tias koj nco qab Pythagorean theorem. Txhua tus tub ntxhais kawm tuaj yeem tau txais cov qauv xav tau. Nws muaj daim ntawv hauv qab no:
D=√(A2+ B2 + C2)
Ntawm no D yog qhov ntev ntawm kab pheeb ces kaum. Cov cim ntxiv yog qhov ntev ntawm ob sab ntawm lub thawv.
Ntau tus neeg tsis meej pem kab kab pheeb ces kaum ntawm parallelepiped nrog kab pheeb ces kaum ntawm nws sab. Hauv qab no yog ib daim duab uas muaj ximcov ntu sawv cev rau kab pheeb ces kaum ntawm ob sab ntawm daim duab.
Qhov ntev ntawm txhua tus kuj tau txiav txim siab los ntawm Pythagorean theorem thiab sib npaug rau lub hauv paus square ntawm cov lej ntawm cov squares ntawm qhov sib txuas sab ntev.
Prism ntim
Ntxiv rau thaj tsam ntawm ib qho quadrangular prism lossis lwm hom prisms, txhawm rau daws qee qhov teeb meem geometric, koj yuav tsum paub lawv qhov ntim. Tus nqi no rau txhua qhov prism yog xam los ntawm cov qauv hauv qab no:
V=Soh
Yog tias lub prism yog duab plaub, ces nws txaus los xam thaj tsam ntawm nws lub hauv paus thiab muab nws los ntawm qhov ntev ntawm ntug ntawm sab kom tau qhov ntim ntawm daim duab.
Yog hais tias lub prism yog ib lub quadrangular prism, ces nws ntim yuav yog:
V=a2h.
Nws yooj yim pom tias cov qauv no hloov mus rau hauv kev qhia rau lub ntim ntawm lub voos xwmfab yog qhov ntev ntawm sab ntug h yog sib npaug rau sab ntawm lub hauv paus a.
Teeb meem nrog cuboid
Txhawm rau muab cov ntaub ntawv kawm tiav, peb yuav daws cov teeb meem hauv qab no: muaj cov duab plaub sib luag uas nws sab yog 3 cm, 4 cm thiab 5 cm.
Rau qhov tseeb, peb yuav xav tias lub hauv paus ntawm daim duab yog ib daim duab plaub nrog ob sab ntawm 3 cm thiab 4 cm. Ces nws cheeb tsam yog 12 cm2, thiab lub sij hawm. yog 14 cm. Siv cov qauv rau thaj tsam ntawm lub prism, peb tau txais:
S=2So+ Sb=212 + 514=24 + 70=94cm2
Txhawm rau txiav txim siab qhov ntev ntawm kab pheeb ces kaum thiab qhov ntim ntawm daim duab, koj tuaj yeem siv cov lus hais saum toj no ncaj qha:
D=√(32+42+52)=7. 071 cm;
V=345=60cm3.
Teeb meem nrog oblique parallelepiped
Daim duab hauv qab no qhia txog oblique prism. Nws sab yog sib npaug: a=10 cm, b=8 cm, c=12 cm. Koj yuav tsum nrhiav thaj chaw ntawm daim duab no.
Ua ntej, cia peb txiav txim siab thaj tsam ntawm lub hauv paus. Daim duab qhia tias lub kaum sab xis yog 50o. Ces nws cheeb tsam yog:
So=ha=sin(50o)ba
Txhawm rau txiav txim siab thaj tsam ntawm thaj chaw sab nraud, koj yuav tsum pom qhov ib puag ncig ntawm cov duab plaub ntxoov ntxoo. Ob sab ntawm daim duab plaub no yog asin(45o) thiab bsin(60o). Tom qab ntawd qhov ib puag ncig ntawm daim duab plaub no yog:
Psr=2(asin(45o)+bsin(60o)
Tag nrho thaj chaw ntawm lub thawv no yog:
S=2So+ Sb=2(sin(50o)ba + acsin(45o) + bcsin(60o)
Peb hloov cov ntaub ntawv los ntawm qhov xwm txheej ntawm qhov teeb meem rau qhov ntev ntawm ob sab ntawm daim duab, peb tau txais cov lus teb:
S=458, 5496 cm3
Nws tuaj yeem pom los ntawm kev daws qhov teeb meem no uas cov haujlwm trigonometric tau siv los txiav txim siab thaj chaw ntawm oblique cov duab.
