Daim duab peb sab pyramid thiab cov qauv rau kev txiav txim siab nws thaj chaw

2024 Tus sau: Angel Austin | [email protected]. Kawg hloov kho: 2023-12-17 05:28

Pyramid yog daim duab geometric spatial, cov yam ntxwv ntawm cov uas tau kawm hauv tsev kawm theem siab hauv chav kawm ntawm cov duab geometry. Hauv kab lus no, peb yuav txiav txim siab txog daim duab peb sab pyramid, nws hom, nrog rau cov qauv rau kev suav nws thaj chaw saum npoo.

Peb tham txog lub pyramid twg?

Ib daim duab peb sab pyramid yog ib daim duab uas tuaj yeem tau txais los ntawm kev sib txuas tag nrho cov vertices ntawm ib daim duab peb sab arbitrary nrog ib qho taw tes uas tsis nyob hauv lub dav hlau ntawm daim duab peb sab. Raws li cov ntsiab lus no, lub pyramid nyob rau hauv kev txiav txim siab yuav tsum muaj qhov pib daim duab peb sab, uas yog hu ua lub hauv paus ntawm daim duab, thiab peb sab daim duab peb sab uas muaj ib sab nrog lub hauv paus thiab txuas rau ib leeg ntawm ib qho chaw. Lub tom kawg yog hu ua sab saum toj ntawm lub pyramid.

Daim duab saum toj no qhia tau hais tias tus kheej kheej kheej kheej kheej kheej.

Daim duab raws li kev txiav txim siab tuaj yeem yog oblique lossis ncaj. Nyob rau hauv tas li ntawd, lub perpendicular poob los ntawm sab saum toj ntawm lub pyramid mus rau nws lub hauv paus yuav tsum sib tshuam nws nyob rau hauv lub geometric center. qhov chaw geometric ntawm ib qho twgdaim duab peb sab yog lub ntsiab lus ntawm kev sib tshuam ntawm nws cov nruab nrab. Qhov chaw geometric coincides nrog qhov nruab nrab ntawm qhov loj ntawm daim duab hauv physics.

Yog tias daim duab peb sab tsis tu ncua (sib npaug) nyob ntawm lub hauv paus ntawm lub pyramid ncaj, ces nws yog hu ua daim duab peb sab. Nyob rau hauv ib lub pyramid tsis tu ncua, txhua sab yog sib npaug rau ib leeg thiab yog daim duab peb sab sib npaug.

Yog tias qhov siab ntawm lub pyramid ib txwm muaj xws li nws cov duab peb sab sab ua qhov sib npaug, ces nws yog hu ua tetrahedron. Nyob rau hauv ib lub tetrahedron, tag nrho plaub lub ntsej muag sib npaug rau ib leeg, yog li txhua tus ntawm lawv tuaj yeem suav tias yog lub hauv paus.

Pyramid ntsiab

Cov ntsiab lus no suav nrog lub ntsej muag lossis ob sab ntawm daim duab, nws cov npoo, ntsug, qhov siab thiab apothems.

Raws li qhia, txhua sab ntawm daim duab peb sab pyramid yog daim duab peb sab. Lawv tus lej yog 4 (3 sab thiab ib qho ntawm lub hauv paus).

Cov kab ntsug yog cov ntsiab lus sib tshuam ntawm peb sab peb sab. Nws tsis yog qhov nyuaj los twv tias rau lub pyramid nyob rau hauv kev txiav txim siab muaj 4 ntawm lawv (3 nyob rau hauv lub hauv paus thiab 1 mus rau sab saum toj ntawm lub pyramid).

Ntug tuaj yeem txhais tau tias yog kab uas hla ob sab peb sab, lossis raws li cov kab uas txuas txhua ob qhov vertices. Tus naj npawb ntawm cov npoo sib raug rau ob zaug ntawm cov hauv paus vertices, uas yog, rau daim duab peb sab pyramid nws yog 6 (3 npoo ntawm lub hauv paus thiab 3 npoo yog tsim los ntawm sab ntsej muag).

Qhov siab, raws li tau sau tseg saum toj no, yog qhov ntev ntawm qhov txuas txuas los ntawm sab saum toj ntawm lub pyramid mus rau nws lub hauv paus. Yog tias peb kos qhov siab ntawm qhov vertex mus rau txhua sab ntawm daim duab peb sab,ces lawv yuav hu ua apotems (los yog apothems). Yog li, daim duab peb sab pyramid muaj ib qhov siab thiab peb apothems. Cov tom kawg yog sib npaug rau ib leeg rau lub pyramid tsis tu ncua.

Lub hauv paus ntawm lub pyramid thiab nws thaj chaw

Vim lub hauv paus rau daim duab hauv qab kev txiav txim siab feem ntau yog daim duab peb sab, txhawm rau xam nws thaj chaw nws txaus los nrhiav nws qhov siab h_o thiab qhov ntev ntawm sab ntawm lub hauv paus a, uas nws yog qis. Tus qauv rau thaj tsam S_o ntawm lub hauv paus yog:

S_o=1/2h_oa

Yog tias daim duab peb sab ntawm lub hauv paus yog qhov sib npaug, ces thaj tsam ntawm lub hauv paus ntawm daim duab peb sab pyramid yog suav nrog cov qauv hauv qab no:

S_o=√3/4a²

Ntawd yog, thaj tsam S_oyog qhov tshwj xeeb txiav txim siab los ntawm qhov ntev ntawm ib sab ntawm daim duab peb sab.

Sab thiab tag nrho cheeb tsam ntawm daim duab

Ua ntej txiav txim siab thaj tsam ntawm daim duab peb sab, nws yog qhov tsim nyog los qhia nws txoj kev loj hlob. Nws yog duab hauv qab no.

Thaj chaw ntawm qhov cheb no tsim los ntawm plaub daim duab peb sab yog tag nrho thaj tsam ntawm lub pyramid. Ib qho ntawm cov duab peb sab sib raug rau lub hauv paus, cov qauv rau kev txiav txim siab tus nqi uas tau sau saum toj no. Peb lub ntsej muag daim duab peb sab ua ke ua rau thaj tsam ntawm daim duab. Yog li ntawd, txhawm rau txiav txim siab tus nqi no, nws txaus los siv cov qauv saum toj no rau daim duab peb sab arbitrary rau txhua tus ntawm lawv, thiab tom qab ntawd ntxiv peb qhov txiaj ntsig.

Yog tias lub pyramid yog lawm, ces qhov kev suavlateral deg cheeb tsam yog yooj yim, vim hais tias tag nrho cov ntsej muag lateral yog zoo tib yam equilateral daim duab peb sab. Denote h_b qhov ntev ntawm lub apothem, ces thaj tsam ntawm sab nraud S_b tuaj yeem txiav txim tau raws li hauv qab no:

S_b=3/2ah_b

Cov qauv no ua raws li cov lus qhia dav dav rau thaj tsam ntawm daim duab peb sab. Tus lej 3 tau tshwm sim hauv cov lej vim qhov tseeb tias lub pyramid muaj peb sab ntsej muag.

Apotema h_b hauv lub pyramid tsis tu ncua tuaj yeem suav tau yog tias qhov siab ntawm daim duab h paub. Kev siv Pythagorean theorem, peb tau txais:

h_b=√(h²+ a²/12)

Pom tseeb, tag nrho cheeb tsam S ntawm daim duab saum npoo yog sib npaug rau cov lej ntawm nws sab thiab hauv paus cheeb tsam:

S=S_o+ S_b

Rau lub pyramid tsis tu ncua, hloov pauv txhua yam kev paub, peb tau txais cov qauv:

S=√3/4a²+ 3/2a√(h²+ a ²/12)

Thaj chaw ntawm daim duab peb sab yog nyob ntawm qhov ntev ntawm sab ntawm nws lub hauv paus thiab qhov siab.

Example problem

Nws paub tias sab ntug ntawm daim duab peb sab yog 7 cm, thiab sab ntawm lub hauv paus yog 5 cm. Koj yuav tsum nrhiav thaj chaw ntawm daim duab yog tias koj paub tias lub pyramid yog ib txwm.

Siv kev sib npaug dav dav:

S=S_o+ S_b

Area S_ois equal to:

S_o=√3/4a² =√3/45^{2 ≈10, 825cm}2.

txhawm rau txiav txim siab thaj tsam sab nraud, koj yuav tsum nrhiav qhov apotema. Nws tsis yog qhov nyuaj los qhia tias dhau ntawm qhov ntev ntawm sab ntug a_b nws yog txiav txim los ntawm tus qauv:

h_b=√(a_b²- a^{2 /4)=√(7}2^{- 5}2^{/4) ≈ 6.538 cm.}

Tom qab ntawd thaj tsam S_b yog:

S_b=3/2ah_b=3/256, 538=49.035 cm2.

Tag nrho cheeb tsam ntawm lub pyramid yog:

S=S_o+ S_b=10.825 + 49.035=59.86cm².

Nco ntsoov tias thaum daws qhov teeb meem, peb tsis tau siv tus nqi ntawm lub pyramid qhov siab hauv kev suav.