Hexagonal prism thiab nws cov yam ntxwv tseem ceeb

2024 Tus sau: Angel Austin | [email protected]. Kawg hloov kho: 2024-01-02 17:48

Spatial geometry yog kev kawm ntawm prisms. Lawv cov yam ntxwv tseem ceeb yog qhov ntim muaj nyob hauv lawv, thaj chaw saum npoo thiab tus naj npawb ntawm cov khoom siv. Hauv kab lus, peb yuav txiav txim siab tag nrho cov khoom no rau lub hexagonal prism.

Peb tham txog dab tsi?

Ib hexagonal prism yog ib daim duab tsim los ntawm ob lub polygons nrog rau rau sab thiab rau lub kaum sab xis, thiab rau parallelograms txuas cov cim hexagons rau hauv ib qho geometric tsim.

Daim duab qhia ib qho piv txwv ntawm no prism.

Lub hexagon cim liab yog hu ua lub hauv paus ntawm daim duab. Obviously, tus naj npawb ntawm nws cov hauv paus yog sib npaug rau ob, thiab ob qho tib si yog tib yam. Lub ntsej muag daj-ntsuab ntawm lub prism hu ua nws sab. Hauv daim duab lawv sawv cev los ntawm cov squares, tab sis feem ntau lawv yog parallelograms.

Lub hexagonal prism tuaj yeem ua tau ncaj thiab ncaj. Hauv thawj kis, lub kaum sab xis ntawm lub hauv paus thiab ob sab tsis ncaj, hauv qhov thib ob lawv sib npaug rau 90o. Tsis tas li ntawd, qhov prism no tuaj yeem raug thiab tsis raug. Tsis tu ncua hexagonallub prism yuav tsum ncaj thiab muaj ib tug tsis tu ncua hexagon ntawm lub hauv paus. Cov prism saum toj no hauv daim duab txaus siab rau cov kev cai no, yog li nws yog hu ua qhov tseeb. Ntxiv rau hauv tsab xov xwm peb yuav kawm tsuas yog nws cov khoom, raws li cov ntaub ntawv dav dav.

Elements

Rau txhua qhov prism nws cov ntsiab lus tseem ceeb yog cov npoo, ntsej muag thiab vertices. Lub hexagonal prism yog tsis muaj kev zam. Daim duab saum toj no tso cai rau koj suav cov lej ntawm cov ntsiab lus no. Yog li, peb tau txais 8 lub ntsej muag lossis ob sab (ob lub hauv paus thiab rau qhov sib npaug rau sab), tus naj npawb ntawm vertices yog 12 (6 vertices rau txhua lub hauv paus), tus naj npawb ntawm cov npoo ntawm hexagonal prism yog 18 (rau sab thiab 12 rau lub hauv paus).

Nyob rau xyoo 1750, Leonhard Euler (tus kws ua lej Swiss) tsim rau txhua tus polyhedra, uas suav nrog prism, kev sib raug zoo ntawm lej ntawm cov lej ntawm cov ntsiab lus qhia. Txoj kev sib raug zoo no zoo li:

ntau tus ntug=tus naj npawb ntawm lub ntsej muag + tus lej ntawm qhov tseem ceeb - 2.

Cov duab saum toj no txaus siab rau cov qauv no.

Prism kab pheeb ces kaum

Txhua kab pheeb ces kaum ntawm hexagonal prism tuaj yeem muab faib ua ob hom:

• cov uas dag hauv lub dav hlau ntawm nws lub ntsej muag;
• cov uas muaj tag nrho cov ntim ntawm daim duab.

Daim duab hauv qab no qhia tag nrho cov kab pheeb suab no.

Yuav pom tau tias D₁ yog sab kab pheeb ces kaum, D₂ thiab D₃ yog lub diagonals tag nrho prism, D₄ thiab D_{5 - kab pheeb ces kaum ntawm lub hauv paus.}

Qhov ntev ntawm kab pheeb ces kaum ntawm ob sab yog sib npaug. Nws yog ib qho yooj yim los xam lawv siv lub npe nrov Pythagorean theorem. Cia ib tug ntev ntawm sab ntawm lub hexagon, b qhov ntev ntawm sab ntug. Tom qab ntawd tus kab pheeb ces kaum muaj qhov ntev:

D₁=√(a² + b^2).

Diagonal D₄ kuj yooj yim los txiav txim. Yog tias peb nco qab tias ib lub hexagon tsis tu ncua haum rau hauv lub voj voog nrog lub vojvoog a, ces D _{4yog txoj kab uas hla ntawm lub voj voog, uas yog, peb tau txais cov qauv hauv qab no:}

D_4=2a.

Diagonal D₅bases nyuaj dua nrhiav. Txhawm rau ua qhov no, xav txog daim duab peb sab sib npaug ABC (saib daim duab). Rau nws AB=BC=a, lub kaum sab xis ABC yog 120^{o. Yog tias peb txo qhov siab ntawm lub kaum sab xis no (nws tseem yuav yog lub bisector thiab nruab nrab), ces ib nrab ntawm AC puag yuav muab sib npaug rau:}

AC/2=ABsin(60o)=a√3/2.

AC sab yog kab pheeb ces kaum ntawm D_{5, yog li peb tau txais:}

D_5=AC=√3a.

Tam sim no nws tseem pom cov kab pheeb ces kaum D₂thiab D_{3 ntawm ib txwm hexagonal prism. Txhawm rau ua qhov no, koj yuav tsum pom tias lawv yog cov hypotenuses ntawm cov duab peb sab sib raug zoo. Siv Pythagorean theorem, peb tau txais:}

D₂=√(D₄²+ b^2)=√(4a2^{+ b}2^);

D₃=√(D₅²+ b^2)=√(3a2^{+ b}2^).

Yog li, kab pheeb ces kaum loj tshaj plaws rau txhua qhov txiaj ntsig ntawm a thiab b yogD_2.

Ntog cheeb tsam

Yuav kom nkag siab txog dab tsi ntawm ceg txheem ntseeg, txoj hauv kev yooj yim tshaj plaws yog xav txog kev txhim kho ntawm qhov prism no. Nws muaj nyob rau hauv daim duab.

Nws tuaj yeem pom tau tias txhawm rau txiav txim siab thaj tsam ntawm txhua sab ntawm daim duab raws li kev txiav txim siab, nws yog ib qho tsim nyog los suav thaj tsam ntawm plaub fab thiab thaj tsam ntawm lub hexagon cais, ces muab faib rau lawv. los ntawm tus lej sib npaug sib npaug rau tus naj npawb ntawm txhua n-gon hauv prism, thiab ntxiv cov txiaj ntsig. Hexagons 2, duab plaub 6.

Rau thaj tsam ntawm ib daim duab peb tau txais:

S_1=ab.

Tom qab ntawd thaj chaw sab nraud yog:

S_2=6ab.

Txhawm rau txiav txim siab thaj tsam ntawm hexagon, txoj hauv kev yooj yim tshaj plaws yog siv cov qauv sib xws, uas zoo li:

S=n/4a2ctg(pi/n).

Hloov tus lej n sib npaug rau 6 rau hauv qhov kev qhia no, peb tau txais thaj tsam ntawm ib hexagon:

S₆=6/4a²ctg(pi/6)=3√3/2a 2^.

Qhov kev qhia no yuav tsum muab faib ua ob kom tau qhov chaw ntawm lub hauv paus ntawm lub prism:

S_os=3√3a^2.

Nws tseem yuav ntxiv S_os thiab S_{2 kom tau tag nrho thaj chaw ntawm daim duab:}

S=S_os+ S₂=3√3a^{2+ 6ab=3a(√3a+2b).}

Prism ntim

Tom qab tus qauv rauthaj tsam ntawm lub hauv paus hexagonal, xam qhov ntim muaj nyob rau hauv lub prism hauv nqe lus nug yog yooj yim li shelling pears. Ua li no, koj tsuas yog yuav tsum tau muab cov cheeb tsam ntawm lub hauv paus pob txha (hexagon) los ntawm qhov siab ntawm daim duab, qhov ntev ntawm qhov sib npaug ntawm qhov ntev ntawm sab ntug. Peb tau txais cov qauv:

V=S₆b=3√3/2a^2b.

Nco ntsoov tias cov khoom ntawm lub hauv paus thiab qhov siab muab tus nqi ntawm qhov ntim ntawm txhua yam prism, suav nrog oblique ib qho. Txawm li cas los xij, nyob rau hauv rooj plaub tom kawg, kev suav ntawm qhov siab yog qhov nyuaj, vim nws yuav tsis sib npaug ntawm qhov ntev ntawm sab tav. Raws li rau ib txwm hexagonal prism, tus nqi ntawm nws qhov ntim yog ib qho kev ua haujlwm ntawm ob qhov sib txawv: sab a thiab b.