Daim duab peb sab, square, hexagon - cov duab no paub rau txhua tus. Tab sis tsis yog txhua leej txhua tus paub dab tsi ib tug niaj zaus polygon yog. Tab sis cov no yog tag nrho cov tib geometric duab. Lub polygon tsis tu ncua yog ib qho uas muaj cov kaum sib npaug thiab sab. Muaj ntau cov duab zoo li no, tab sis lawv txhua tus muaj cov khoom qub, thiab cov qauv tib yam siv rau lawv.
Properties of regular polygons
Txhua tus polygon tsis tu ncua, yog nws lub square lossis octagon, tuaj yeem sau rau hauv lub voj voog. Cov cuab yeej yooj yim no feem ntau siv thaum tsim ib daim duab. Tsis tas li ntawd, lub voj voog tseem tuaj yeem sau rau hauv ib lub voj voog. Nyob rau hauv cov ntaub ntawv no, tus naj npawb ntawm cov ntsiab lus ntawm kev sib cuag yuav sib npaug rau tus naj npawb ntawm nws sab. Nws yog ib qho tseem ceeb uas lub voj voog inscribed nyob rau hauv ib tug niaj zaus polygon yuav muaj ib qho chaw nrog nws. Cov duab geometric no raug rau tib lub theorems. Ib sabntawm ib txwm n-gon muaj feem xyuam rau lub vojvoog R ntawm lub voj voog ncig nws, yog li, nws tuaj yeem suav nrog cov qauv hauv qab no: a=2R ∙ sin180 °. Los ntawm lub vojvoog ntawm lub voj voog, koj tuaj yeem pom tsis tau tsuas yog sab, tab sis kuj nyob ib puag ncig ntawm lub voj voog.
Yuav ua li cas thiaj nrhiav tau tus naj npawb ntawm sab ntawm tus lej tsis tu ncua
Ib yam li niaj zaus n-gon muaj qee qhov sib npaug ntawm ib leeg, uas, thaum sib txuas, tsim kab kaw. Nyob rau hauv cov ntaub ntawv no, tag nrho cov ces kaum ntawm tus tsim daim duab muaj ib yam nqi. Polygons muab faib ua yooj yim thiab complex. Thawj pawg muaj xws li daim duab peb sab thiab ib lub xwmfab. Complex polygons muaj ntau sab. Lawv kuj suav nrog cov duab hnub qub. Rau txoj kev tsis tu ncua polygons, sab yog pom los ntawm kev sau lawv hauv ib lub voj voog. Cia peb muab pov thawj. Kos ib tug tsis tu ncua polygon nrog tus naj npawb ntawm sab n. Piav ib lub voj voog ncig nws. Qhia lub vojvoog R. Tam sim no xav txog tias qee qhov n-gon tau muab. Yog hais tias cov ntsiab lus ntawm nws cov ces kaum pw ntawm ib lub voj voog thiab sib npaug rau ib leeg, ces ob sab tuaj yeem pom los ntawm tus qauv: a=2R ∙ sinα: 2.
Nrhiav tus naj npawb ntawm sab ntawm daim duab peb sab inscribed ib txwm
Ib daim duab peb sab sib npaug yog ib qho polygon tsis tu ncua. Tib cov qauv siv rau nws raws li lub square thiab n-gon. Daim duab peb sab yuav raug suav hais tias yog qhov tseeb yog tias nws muaj qhov ntev tib yam. Hauv qhov no, cov ces kaum yog 60⁰. Tsim ib daim duab peb sab nrog rau sab ntev a. Paub nws qhov nruab nrab thiab qhov siab,koj tuaj yeem nrhiav tus nqi ntawm nws sab. Txhawm rau ua qhov no, peb yuav siv txoj hauv kev nrhiav los ntawm tus qauv a \u003d x: cosα, qhov twg x yog qhov nruab nrab lossis qhov siab. Txij li txhua sab ntawm daim duab peb sab sib npaug, peb tau txais a=b=c. Ces cov lus hauv qab no yuav muaj tseeb a=b=c=x: cosα. Ib yam li ntawd, koj tuaj yeem pom tus nqi ntawm ob sab hauv daim duab peb sab isosceles, tab sis x yuav yog qhov siab muab. Nyob rau tib lub sijhawm, nws yuav tsum tau npaj nruj me ntsis ntawm lub hauv paus ntawm daim duab. Yog li, paub qhov siab x, peb pom ib sab ntawm daim duab peb sab isosceles siv cov qauv a \u003d b \u003d x: cosα. Tom qab nrhiav tus nqi ntawm a, koj tuaj yeem xam qhov ntev ntawm lub hauv paus c. Cia peb siv Pythagorean theorem. Peb yuav nrhiav tus nqi ntawm ib nrab ntawm lub hauv paus c: 2=√(x: cosα)^2 - (x^2)=√x^2 (1 - cos^2α): cos^2α=x ∙ tgα. Ces c=2xtanα. Nov yog ib txoj hauv kev yooj yim los nrhiav cov naj npawb ntawm sab ntawm ib qho kev sau ntau ntau.
Xaiv ob sab ntawm lub square sau nyob rau hauv ib lub voj voog
Zoo li lwm yam sau ntau yam tsis tu ncua, ib lub xwmfab muaj ob sab thiab kaum sib npaug. Tib cov qauv siv rau nws raws li daim duab peb sab. Koj tuaj yeem xam cov sab ntawm ib lub square siv tus nqi ntawm kab pheeb ces kaum. Cia peb xav txog txoj kev no kom ntxaws ntxiv. Nws paub tias kab pheeb ces kaum bisects lub kaum sab xis. Thaum xub thawj, nws tus nqi yog 90 degrees. Yog li, tom qab kev faib, ob lub kaum sab xis yog tsim. Lawv lub kaum sab xis yuav yog 45 degrees. Raws li, txhua sab ntawm lub square yuav sib npaug, uas yog: a \u003d c \u003d c \u003d d \u003d e ∙ cosα \u003d e √ 2: 2, qhov twg e yog kab pheeb ces kaum ntawm lub square, los yog lub hauv paus ntawm txoj cai daim duab peb sab tsim tom qab faib. Nws tsis yog tib txoj kevnrhiav ob sab ntawm ib lub square. Cia peb sau daim duab no rau hauv ib lub voj voog. Paub lub vojvoog ntawm lub voj voog R no, peb pom sab ntawm lub square. Peb yuav xam nws raws li nram no a4=R√2. Cov radii ntawm cov polygons niaj zaus yog xam los ntawm cov qauv R=a: 2tg (360o: 2n), qhov twg a yog qhov ntev.
Yuav ua li cas xam qhov ib puag ncig ntawm n-gon
Lub cheeb tsam ntawm ib qho n-gon yog qhov suav ntawm txhua sab. Nws yog ib qho yooj yim los xam nws. Ua li no, koj yuav tsum paub qhov tseem ceeb ntawm txhua sab. Rau qee hom polygons, muaj cov qauv tshwj xeeb. Lawv tso cai rau koj mus nrhiav tau lub perimeter sai dua. Nws paub tias txhua tus polygon tsis tu ncua muaj qhov sib npaug. Yog li ntawd, txhawm rau xam nws thaj tsam, nws yog txaus kom paub tsawg kawg yog ib qho ntawm lawv. Cov mis yuav nyob ntawm seb muaj pes tsawg tus sab ntawm daim duab. Feem ntau, nws zoo li qhov no: P \u003d ib qho, qhov twg a yog tus nqi ntawm sab, thiab n yog cov kaum sab xis. Piv txwv li, txhawm rau nrhiav qhov ib puag ncig ntawm lub octagon tsis tu ncua nrog ib sab ntawm 3 cm, koj yuav tsum muab nws los ntawm 8, uas yog, P=3 ∙ 8=24 cm. Rau ib hexagon nrog ib sab ntawm 5 cm, peb xam. raws li nram no: P=5 ∙ 6=30 cm.
Nrhiav qhov ib puag ncig ntawm qhov parallelogram, square thiab rhombus
Nyob ntawm seb muaj pes tsawg sab ib lub polygon ib txwm muaj, nws qhov ib puag ncig raug suav. Qhov no ua rau txoj haujlwm yooj yim dua. Tseeb, tsis zoo li lwm cov duab, nyob rau hauv cov ntaub ntawv no nws tsis yog yuav tsum tau mus nrhiav rau tag nrho nws sab, tsuas yog ib tug txaus. Los ntawm tib txoj cai, peb pom lub perimeter ntawmquadrangles, uas yog, square thiab rhombus. Txawm tias muaj tseeb tias cov no yog cov duab sib txawv, cov qauv rau lawv yog tib yam P=4a, qhov twg a yog sab. Cia peb ua piv txwv. Yog hais tias ib sab ntawm lub rhombus los yog square yog 6 cm, ces peb pom lub perimeter raws li nram no: P \u003d 4 ∙ 6 \u003d 24 cm. Ib tug parallelogram tsuas muaj opposite sab. Yog li ntawd, nws thaj tsam yog pom siv ib txoj kev sib txawv. Yog li, peb yuav tsum paub qhov ntev a thiab qhov dav b ntawm daim duab. Tom qab ntawd peb siv cov qauv P=(a + c) ∙ 2. Ib qho parallelogram, uas txhua sab thiab cov ces kaum ntawm lawv sib npaug, hu ua rhombus.
Nrhiav qhov ib puag ncig ntawm ib daim duab peb sab thiab txoj cai
Qhov ib puag ncig ntawm daim duab peb sab sib npaug tuaj yeem pom los ntawm tus qauv P=3a, qhov twg a yog qhov ntev ntawm sab. Yog tias nws tsis paub, nws tuaj yeem pom los ntawm qhov nruab nrab. Hauv daim duab peb sab txoj cai, tsuas yog ob sab sib npaug. Lub hauv paus tuaj yeem pom los ntawm Pythagorean theorem. Tom qab cov txiaj ntsig ntawm tag nrho peb sab dhau los paub, peb suav cov perimeter. Nws tuaj yeem pom los ntawm kev siv cov mis P \u003d a + b + c, qhov twg a thiab b yog sib npaug, thiab c yog lub hauv paus. Nco qab tias hauv daim duab peb sab isosceles a \u003d b \u003d a, yog li ntawd, a + b \u003d 2a, ces P \u003d 2a + c. Piv txwv li, sab ntawm daim duab peb sab isosceles yog 4 cm, nrhiav nws lub hauv paus thiab ib puag ncig. Peb xam tus nqi ntawm lub hypotenuse siv Pythagorean theorem c=√a2 + v2=√16 + 16=√32=5.65 cm. Tam sim no peb xam perimeter Р=2 ∙ 4 + 5, 65=13.65 cm.
Yuav ua li cas thiaj nrhiav tau cov ces kaum ntawm ib lub polygon tsis tu ncua
Regular polygontshwm sim nyob rau hauv peb lub neej txhua hnub, piv txwv li, ib tug dog dig square, daim duab peb sab, octagon. Nws yuav zoo li tsis muaj dab tsi yooj yim dua li tsim daim duab no koj tus kheej. Tab sis qhov no tsuas yog thaum xub thawj siab ib muag. Txhawm rau tsim ib qho n-gon, koj yuav tsum paub tus nqi ntawm nws cov ces kaum. Tab sis koj yuav nrhiav tau lawv li cas? Txawm tias cov kws tshawb fawb ntawm ancient sij hawm sim tsim cov polygons tsis tu ncua. Lawv kwv yees kom haum lawv rau hauv lub voj voog. Thiab tom qab ntawd cov ntsiab lus tsim nyog tau cim rau ntawm nws, txuas nrog cov kab ncaj nraim. Rau cov duab yooj yim, qhov teeb meem kev tsim kho tau raug daws. Cov qauv thiab theorems tau txais. Piv txwv li, Euclid hauv nws txoj haujlwm nto moo "The Beginning" tau koom nrog kev daws teeb meem rau 3-, 4-, 5-, 6- thiab 15-gons. Nws nrhiav tau txoj hauv kev los tsim lawv thiab nrhiav cov ces kaum. Cia peb saib yuav ua li cas rau qhov no rau 15-gon. Ua ntej koj yuav tsum xam cov lej ntawm nws cov kaum sab xis. Nws yog ib qho tsim nyog yuav tau siv cov qauv S=180⁰(n-2). Yog li, peb muab 15-gon, uas txhais tau hais tias tus lej n yog 15. Peb hloov cov ntaub ntawv peb paub rau hauv cov qauv thiab tau S=180⁰ (15 - 2)=180⁰ x 13=2340⁰. Peb tau pom cov lej ntawm tag nrho cov kaum sab hauv ntawm 15-gon. Tam sim no peb yuav tsum tau txais tus nqi ntawm txhua tus ntawm lawv. Muaj 15 lub kaum ntse ntse, peb ua qhov suav 2340⁰: 15=156⁰. Qhov no txhais tau hais tias txhua lub kaum sab hauv yog 156⁰, tam sim no siv tus pas ntsuas thiab lub compass, koj tuaj yeem tsim 15-gon li niaj zaus. Tab sis ua li cas txog ntau txoj kev n-gons? Tau ntau pua xyoo, cov kws tshawb fawb tau tawm tsam los daws qhov teeb meem no. Nws tsuas yog pom nyob rau xyoo pua 18th los ntawm Carl Friedrich Gauss. Nws muaj peev xwm tsim tau 65537-gon. Txij thaum ntawd los, qhov teeb meem raug txiav txim siab daws tiav.
Xaiv kaum ntawm n-gonshauv radians
Tau kawg, muaj ob peb txoj hauv kev los nrhiav cov ces kaum ntawm polygons. Feem ntau lawv yog xam nyob rau hauv degrees. Tab sis koj tuaj yeem qhia lawv hauv radians. Yuav ua li cas? Nws yog ib qho tsim nyog yuav tau ua raws li hauv qab no. Ua ntej, peb pom tus naj npawb ntawm sab ntawm ib lub polygon tsis tu ncua, ces rho tawm 2 los ntawm nws. Yog li, peb tau txais tus nqi: n - 2. Muab qhov sib txawv ntawm tus lej n ("pi"=3, 14). Tam sim no nws tseem tsuas yog faib cov khoom tsim los ntawm tus naj npawb ntawm cov ces kaum hauv n-gon. Xav txog cov kev suav no siv cov piv txwv ntawm tib lub kaum tsib-sab. Yog li, tus lej n yog 15. Siv cov qauv S=p(n - 2): n=3, 14(15 - 2): 15=3, 14 ∙ 13: 15=2, 72. Qhov no, tau kawg, tsis yog tib txoj kev los xam lub kaum sab xis hauv radians. Koj tuaj yeem faib qhov loj ntawm lub kaum sab xis hauv degrees los ntawm tus lej 57, 3. Tom qab tag nrho, qhov ntau degrees yog sib npaug rau ib radian.
Xaiv tus nqi ntawm cov kaum hauv degrees
Dhau li ntawm cov degrees thiab radians, koj tuaj yeem sim nrhiav tus nqi ntawm cov kaum sab xis ntawm tus lej tsis tu ncua hauv qib. Qhov no yog ua raws li hauv qab no. Tshem tawm 2 los ntawm tag nrho cov kaum sab xis, faib qhov sib txawv ntawm qhov sib txawv los ntawm tus naj npawb ntawm sab ntawm ib qho polygon tsis tu ncua. Peb muab qhov tshwm sim los ntawm 200. Los ntawm txoj kev, xws li ib chav ntsuas ntawm cov ces kaum li hailstones yog xyaum tsis siv.
Xaiv cov kaum sab nraud ntawm n-gons
Rau txhua qhov ntau npaum li cas, tshwj tsis yog rau sab hauv, koj tuaj yeem suav lub kaum sab nraud. Nws tus nqi yog pom nyob rau hauv tib txoj kev raws li rau lwm cov duab. Yog li, txhawm rau nrhiav lub kaum sab nrauv ntawm lub polygon tsis tu ncua, koj xav taupaub lub ntsiab lus ntawm sab hauv. Tsis tas li ntawd, peb paub tias qhov sib npaug ntawm ob lub kaum ntse ntse yog 180 degrees. Yog li ntawd, peb ua cov kev xam raws li nram no: 180⁰ rho tawm tus nqi ntawm lub sab hauv lub kaum sab xis. Peb pom qhov txawv. Nws yuav muab sib npaug rau tus nqi ntawm lub kaum sab xis nyob ib sab ntawm nws. Piv txwv li, lub kaum sab hauv ntawm ib lub xwmfab yog 90 °, yog li lub kaum sab nrauv yuav yog 180⁰ - 90⁰=90⁰. Raws li peb pom, nws tsis yooj yim nrhiav nws. Lub kaum sab nraud tuaj yeem coj tus nqi ntawm +180⁰ txog -180⁰, feem.
