Thaum npaj rau kev xeem hauv lej, cov tub ntxhais kawm yuav tsum ua kom lawv txoj kev paub txog algebra thiab geometry. Kuv xav muab tag nrho cov ntaub ntawv paub, piv txwv li, yuav ua li cas xam thaj tsam ntawm lub pyramid. Ntxiv mus, pib los ntawm lub hauv paus thiab sab ntsej muag mus rau tag nrho cov saum npoo. Yog tias qhov xwm txheej pom tseeb nrog sab ntsej muag, vim lawv yog daim duab peb sab, ces lub hauv paus yeej ib txwm txawv.
Yuav ua li cas nrhiav thaj chaw ntawm lub hauv paus ntawm lub pyramid?
Nws tuaj yeem yog txhua yam zoo li: los ntawm daim duab peb sab arbitrary mus rau n-gon. Thiab lub hauv paus no, ntxiv rau qhov sib txawv ntawm cov kaum sab xis, tuaj yeem ua ib daim duab tsis tu ncua lossis ib qho tsis raug. Hauv USE cov haujlwm uas txaus siab rau cov menyuam kawm ntawv, tsuas yog cov haujlwm uas muaj cov lej raug ntawm lub hauv paus. Yog li ntawd, peb tsuas tham txog lawv.
Daim duab peb sab
Nws yog qhov sib npaug. Ib qho uas txhua sab sib npaug thiab txhais los ntawm tsab ntawv "a". Hauv qhov no, thaj tsam ntawm lub hauv paus ntawm lub pyramid yog xam los ntawm cov qauv:
S=(a2√3) / 4.
Square
Cov mis rau xam nws thaj chaw yog qhov yooj yim tshaj plaws,ntawm no "a" yog sab dua:
S=a2.
Txoj kev tsis tu ncua n-gon
Sab ntawm ib tug polygon muaj tib lub npe. Rau tus naj npawb ntawm cov ces kaum, tsab ntawv Latin n yog siv.
S=(na2) / (4tg (180º/n)).
Yuav ua li cas xam lateral thiab tag nrho cheeb tsam?
Vim lub hauv paus yog daim duab tsis tu ncua, txhua sab ntawm lub pyramid sib npaug. Tsis tas li ntawd, txhua tus ntawm lawv yog daim duab peb sab isosceles, vim cov npoo sab sib npaug. Tom qab ntawd, txhawm rau txhawm rau suav thaj tsam sab nraud ntawm pyramid, koj xav tau cov qauv uas muaj cov lej ntawm cov monomials zoo tib yam. Tus naj npawb ntawm cov ntsiab lus yog txiav txim los ntawm tus naj npawb ntawm sab ntawm lub hauv paus.
Qhov cheeb tsam ntawm daim duab peb sab isosceles yog xam los ntawm cov qauv uas ib nrab ntawm cov khoom ntawm lub hauv paus yog muab los ntawm qhov siab. Qhov siab hauv lub pyramid no hu ua apothem. Nws lub npe yog "A". Cov qauv dav dav rau thaj chaw sab nraud yog:
S=½ PA, qhov twg P yog thaj tsam ntawm lub hauv paus ntawm lub pyramid.
Muaj cov xwm txheej thaum ob sab ntawm lub hauv paus tsis paub, tab sis sab npoo (c) thiab lub kaum sab xis ntawm nws vertex (α) tau muab. Tom qab ntawd nws yuav tsum siv cov qauv no los xam cov cheeb tsam sab nraud ntawm lub pyramid:
S=n/2in2 sin α.
Problem 1
Zoo. Nrhiav tag nrho cheeb tsam ntawm lub pyramid yog tias nws lub hauv paus yog daim duab peb sab sib npaug nrog ib sab ntawm 4 cm, thiab lub apothem yog √3 cm.
Kev txiav txim siab. NwsKoj yuav tsum tau pib los ntawm kev xam qhov perimeter ntawm lub hauv paus. Txij li qhov no yog daim duab peb sab tsis tu ncua, ces P \u003d 34 \u003d 12 cm. Txij li lub npe hu ua apothem, koj tuaj yeem xam tam sim ntawm thaj tsam ntawm tag nrho sab sab: ½12√3=6 √3 cm 2.
Rau ib daim duab peb sab ntawm lub hauv paus, koj tau txais cov nqi hauv qab no: (42√3) / 4=4√3 cm2.
Txhawm rau txiav txim siab tag nrho cheeb tsam, koj yuav tsum tau ntxiv ob qhov txiaj ntsig: 6√3 + 4√3=10√3 cm2.
teb. 10√3cm2.
Problem 2
Txoj Cai. Muaj ib lub pyramid tsis tu ncua. Qhov ntev ntawm sab ntawm lub hauv paus yog 7 hli, sab ntug yog 16 hli. Koj yuav tsum paub nws thaj chaw saum npoo av.
Kev txiav txim siab. Txij li thaum lub polyhedron yog quadrangular thiab tsis tu ncua, ces nws lub hauv paus yog square. Tau kawm txog thaj chaw ntawm lub hauv paus thiab sab ntsej muag, nws yuav ua tau los xam thaj tsam ntawm lub pyramid. Cov mis rau lub square yog muab saum toj no. Thiab ntawm sab ntsej muag, txhua sab ntawm daim duab peb sab paub. Yog li ntawd, koj tuaj yeem siv Heron tus qauv los xam lawv thaj chaw.
Thawj qhov kev suav yooj yim thiab ua rau tus lej no: 49 mm2. Rau qhov thib ob tus nqi, koj yuav tsum xam lub semi-perimeter: (7 + 162): 2=19.5 mm. Tam sim no koj tuaj yeem xam thaj tsam ntawm daim duab peb sab isosceles: √(19.5(19.5-7)(19.5-16)2)=√2985.9375=54.644 mm 2. Tsuas muaj plaub daim duab peb sab xwb, yog li thaum xam tus lej kawg, koj yuav tsum tau muab faib ua 4.
Nws hloov tawm: 49 + 454, 644=267, 576 mm2.
teb. Desired tus nqi 267,576mm2.
Problem 3
Txoj Cai. Rau ib tug li niaj zaus quadrangular pyramid, koj yuav tsum xam lub cheeb tsam. Nws paub sab ntawm lub square - 6 cm thiab qhov siab - 4 cm.
Kev txiav txim siab. Txoj kev yooj yim tshaj plaws yog siv cov qauv nrog cov khoom ntawm lub voj voog thiab lub apothem. Thawj tus nqi yog qhov yooj yim nrhiav. Qhov thib ob yog qhov nyuaj me ntsis.
Peb yuav tsum nco ntsoov lub Pythagorean theorem thiab xav txog ib daim duab peb sab. Nws yog tsim los ntawm qhov siab ntawm lub pyramid thiab lub apothem, uas yog lub hypotenuse. Qhov thib ob ceg yog sib npaug rau ib nrab ntawm sab ntawm lub square, txij li qhov siab ntawm polyhedron ntog rau hauv nruab nrab.
Qhov xav tau apothem (lub hypotenuse ntawm ib daim duab peb sab) yog √(32 + 42)=5 (cm).
Tam sim no koj tuaj yeem suav tus nqi xav tau: ½(46)5+62=96 (saib2).
teb. 96cm2.
Problem 4
Zoo. Muab ib txwm hexagonal pyramid. Sab ntawm nws lub hauv paus yog 22 hli, sab tav yog 61 hli. Dab tsi yog thaj chaw sab nrauv ntawm lub polyhedron no?
Kev txiav txim siab. Qhov laj thawj hauv nws yog tib yam li tau piav qhia hauv qhov teeb meem No.2. Tsuas yog muaj ib lub pyramid nrog lub square ntawm lub hauv paus, thiab tam sim no nws yog hexagon.
Ua ntej tshaj plaws, thaj tsam ntawm lub hauv paus yog suav nrog cov qauv saum toj no: (6222) / (4tg (180º/6))=726/(tg30º)=726 √3 cm2.
Tam sim no koj yuav tsum paub qhov ib puag ncig ntawm ib daim duab peb sab isosceles, uas yog sab ntsej muag. (22 + 612): 2 \u003d 72 cm. Nws tseem yuav suav cov cheeb tsam ntawm lub puam xws lidaim duab peb sab, thiab mam li muab nws los ntawm rau thiab ntxiv rau qhov uas tig tawm rau lub hauv paus.
Calculation by Heron's formula: √(72(72-22)(72-61)2)=√435600=660 cm2 . Kev suav uas yuav muab rau thaj tsam sab nraud: 6606=3960 cm2. Nws tseem yuav ntxiv rau lawv kom paub txog tag nrho qhov chaw: 5217, 47≈5217 cm2.
teb. Base - 726√3cm2, sab nto - 3960cm2, tag nrho cheeb tsam - 5217cm2.
