Thaj tsam ntawm qhov chaw sab nraud thiab qhov ntim ntawm lub pyramid truncated: cov qauv thiab piv txwv ntawm kev daws qhov teeb meem raug

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

Thaum kawm txog cov khoom ntawm cov duab hauv qhov chaw peb sab hauv lub hauv paus ntawm stereometry, feem ntau yuav tsum tau daws teeb meem los txiav txim qhov ntim thiab qhov chaw. Nyob rau hauv tsab xov xwm no, peb yuav qhia yuav ua li cas xam lub ntim thiab lateral deg cheeb tsam rau ib tug truncated pyramid siv cov qauv paub zoo.

Pyramid hauv geometry

Nyob rau hauv geometry, lub pyramid zoo tib yam yog ib daim duab nyob rau hauv qhov chaw, uas yog ua nyob rau hauv ib co ca n-gon. Tag nrho nws cov vertices txuas mus rau ib qho chaw nyob sab nraum lub dav hlau ntawm lub polygon. Piv txwv li, ntawm no yog ib daim duab uas qhia txog lub pentagonal pyramid.

Daim duab no yog tsim los ntawm lub ntsej muag, ntsug thiab ntug. Lub ntsej muag pentagonal hu ua lub hauv paus. Qhov seem daim duab peb sab ntsej muag tsim rau sab saum toj. Qhov kev sib tshuam ntawm txhua daim duab peb sab yog lub ntsiab vertex ntawm lub pyramid. Yog hais tias ib qho perpendicular raug txo qis los ntawm nws mus rau hauv paus, ces ob txoj kev xaiv rau txoj hauj lwm ntawm txoj kev sib tshuam yog ua tau:

• nyob rau hauv qhov chaw geometric, ces lub pyramid hu ua kab ncaj;
• tsis nyob hauvgeometric center, ces daim duab yuav oblique.

Ntxiv mus peb yuav txiav txim siab tsuas yog cov duab ncaj nraim nrog lub hauv paus n-gonal.

Daim duab no yog dab tsi - lub pyramid truncated?

Txhawm rau txiav txim siab qhov ntim ntawm lub pyramid truncated, nws yog ib qho tsim nyog kom nkag siab meej tias daim duab twg yog cov lus nug tshwj xeeb. Cia peb qhia qhov teeb meem no.

Piv txwv tias peb nqa lub dav hlau txiav uas sib npaug rau lub hauv paus ntawm lub pyramid zoo tib yam thiab txiav tawm ib feem ntawm sab saum npoo nrog nws. Yog tias qhov haujlwm no ua tiav nrog lub pentagonal pyramid uas tau qhia saum toj no, koj yuav tau txais daim duab zoo li hauv daim duab hauv qab no.

Los ntawm daim duab nws tuaj yeem pom tias lub pyramid no twb muaj ob lub hauv paus, thiab sab saum toj zoo ib yam li hauv qab, tab sis nws me dua. Lub lateral nto tsis sawv cev los ntawm peb tog, tab sis los ntawm trapezoids. Lawv yog cov isosceles, thiab lawv cov lej sib raug rau cov naj npawb ntawm sab ntawm lub hauv paus. Daim duab truncated tsis muaj lub ntsiab vertex, zoo li lub pyramid li niaj zaus, thiab nws qhov siab yog txiav txim siab los ntawm qhov kev ncua deb ntawm qhov sib npaug.

Hauv cov ntaub ntawv dav dav, yog tias daim duab nyob rau hauv kev txiav txim siab yog tsim los ntawm n-gonal hauv paus, nws muaj n + 2 lub ntsej muag lossis sab, 2n vertices thiab 3n npoo. Ntawd yog, lub pyramid truncated yog polyhedron.

Formula rau ntim ntawm lub pyramid truncated

Nco ntsoov tias qhov ntim ntawm lub pyramid zoo tib yam yog 1/3 ntawm cov khoom ntawm nws qhov siab thiab hauv paus cheeb tsam. Cov mis no tsis haum rau lub pyramid truncated, vim nws muaj ob lub hauv paus. Thiab nws ntimyuav ib txwm tsawg dua tus nqi qub rau daim duab uas nws tau muab los.

Yog tsis nkag mus rau hauv lej cov ntsiab lus ntawm kev tau txais cov lus qhia, peb nthuav qhia cov qauv kawg rau qhov ntim ntawm lub pyramid truncated. Nws yog sau raws li nram no:

V=1/3h(S₁+ S₂+ √(S₁ S₂₎

Ntawm no S₁ thiab S_{2 yog thaj chaw ntawm lub hauv paus qis thiab sab saud, raws li, h yog qhov siab ntawm daim duab. Cov lus sau tau siv tsis tau tsuas yog rau ib qho ncaj nraim truncated pyramid, tab sis kuj rau ib daim duab ntawm chav kawm no. Ntxiv mus, tsis hais hom ntawm lub hauv paus polygons. Ib qho xwm txheej txwv tsis pub siv cov lus qhia rau V yog qhov xav tau rau cov hauv paus ntawm lub pyramid kom sib npaug rau ib leeg.}

Ntau qhov kev txiav txim siab tseem ceeb tuaj yeem kos los ntawm kev kawm cov khoom ntawm cov qauv no. Yog li, yog tias thaj tsam ntawm lub hauv paus sab saud yog xoom, ces peb tuaj rau tus qauv rau V ntawm lub pyramid zoo tib yam. Yog hais tias cov cheeb tsam ntawm lub hauv paus yog sib npaug zos rau ib leeg, ces peb tau txais cov qauv rau lub ntim ntawm lub prism.

Yuav ua li cas thiaj txiav txim siab thaj tsam sab nraud?

Kev paub txog cov yam ntxwv ntawm lub pyramid truncated yuav tsum tsis tsuas yog lub peev xwm los xam nws cov ntim, tab sis kuj yuav tsum paub yuav ua li cas los txiav txim qhov chaw ntawm lub lateral nto.

Truncated pyramid muaj ob hom ntsej muag:

• isosceles trapezoids;
• polygonal bases.

Yog hais tias muaj ib tug tsis tu ncua polygon nyob rau hauv lub hauv paus, ces kev suav ntawm nws cheeb tsam tsis sawv cev lojnyuaj. Ua li no, koj tsuas yog yuav tsum paub qhov ntev ntawm sab a thiab lawv tus lej n.

Nyob rau hauv cov ntaub ntawv ntawm ib tug lateral nto, kev xam ntawm nws cheeb tsam yuav txiav txim siab tus nqi no rau txhua tus n trapezoids. Yog hais tias tus n-gon yog lawm, ces tus qauv rau lub lateral nto ua:

S_b=h_bn(a₁+a_2)/2

Ntawm no h_b yog qhov siab ntawm lub trapezoid, uas yog hu ua daim duab apoteme. Qhov ntau a₁ thiab_{2yog qhov ntev ntawm ob sab ntawm cov hauv paus n-gonal.}

Rau txhua txhua ntu n-gonal truncated pyramid, apotema h_b tuaj yeem txhais tau tshwj xeeb los ntawm qhov tsis muaj a₁ thiab a _{2thiab qhov siab h ntawm cov duab.}

Txoj haujlwm ntawm kev suav qhov ntim thiab thaj tsam ntawm daim duab

Muab ib daim duab peb sab truncated pyramid tsis tu ncua. Nws paub tias nws qhov siab h yog 10 cm, thiab qhov ntev ntawm ob sab ntawm lub hauv paus yog 5 cm thiab 3 cm. Qhov ntim ntawm lub truncated pyramid yog dab tsi thiab thaj tsam ntawm nws lateral nto?

Ua ntej, cia peb suav tus nqi V. Ua qhov no, nrhiav thaj chaw ntawm cov duab peb sab sib npaug nyob rau ntawm lub hauv paus ntawm daim duab. Peb muaj:

S₁=√3/4a₁²=√3/4 5²=10.825cm^2;

S₂=√3/4a₂²=√3/4 3²=3.897 cm²

Hloov cov ntaub ntawv rau hauv cov qauv rau V, peb tau txais qhov ntim xav tau:

V=1/310(10, 825 + 3, 897 + √(10, 8253, 897)) ≈ 70.72 cm³

Txhim kho sab saum toj, koj yuav tsum paubapothem length h_{b. Xav txog qhov sib npaug ntawm daim duab peb sab hauv lub pyramid, peb tuaj yeem sau qhov sib npaug rau nws:}

h_b=√((√3/6(a₁- a₂))²+ h^{2) ≈ 10.017 cm}

Tus nqi ntawm apothem thiab ob sab ntawm daim duab peb sab yog hloov pauv mus rau qhov kev qhia rau S_{b thiab peb tau txais cov lus teb:}

S_b=h_bn(a₁+a₂)/2=10.0173(5+3)/2 ≈ 120.2cm²

Yog li no, peb teb tag nrho cov lus nug ntawm qhov teeb meem: V ≈ 70.72 cm³, S_b ≈ 120.2 cm2^.