Otukukira kunja polygons. Tanthauzo la polygon otukukira kunja. The diagonals wa polygon otukukira kunja

akalumikidzidwa izi zojambula ali ponseponse. Otukukira kunja polygons chilengedwe, monga chisa cha uchi kapena yokumba (munthu anapanga). Chiwerengerochi ntchito popanga osiyanasiyana zokutira zojambulajambula, zomangamanga, zokongoletsera, etc. polygons otukukira kunja ndi chuma kuti mfundo wawo mbali imodzi ya mzere wolunjika kuti umadutsa ndi awiri mfundo moyandikana wa chithunzi geometrical. Pali matanthauzo ena. Iwo amatchedwa polygon otukukira kunja, amene anakonza mu umodzi theka ndege za mzere aliyense molunjika munali munthu wa m'mbali mwake.

polygons otukukira kunja

Mu njira ya masamu pulayimale nthawizonse ankachitira polygons wophweka kwambiri. Kumvetsa katundu wa akalumikidzidwa zojambula muyenera kudziwa chikhalidwe chawo. Kuti kumvetsa kuti chatsekedwa ndi mzere aliyense amene matsirizidwe ali ofanana. Ndi chithunzi popangidwa ndi izo, akhoza osiyanasiyana masanjidwe. Polygon amatchedwa yosavuta chatsekedwa polyline amene moyandikana mayunitsi si ili pa mzere umodzi wowongoka. Maulalo ake ndi mfundo ndi, motero, mumbali ndi pamwamba pa chithunzi geometrical. A polyline yosavuta sayenera intersect yokha.

mfundo za polygon akuitanidwira oyandikana nawo, Ngati muli kumalekezero a mmodzi wa m'mbali mwake. A chithunzi zojambula lomwe lili N-TH chiwerengero cha mfundo, motero N-TH chiwerengero cha zipani wotchedwa N-gon. Lokha wosweka mzere ndi malire kapena mizere ya chithunzi zojambula. Polygonal ndege kapena mosabisa polygon otchedwa otsiriza ndege iliyonse, zochepa awo. Kubwalo mbali ya chithunzi zojambula wotchedwa zigawo polyline wochokera ku vertex chomwecho. Iwo sadzakhala anansi ngati zochokera mfundo zosiyanasiyana polygon lapansi.

matanthauzo ena a polygons otukukira kunja

Mu Masamu pulayimale, pali zingapo ofanana matanthauzo tanthauzo, zikusonyeza zimene amatchedwa polygon otukukira kunja. Komanso, mawu onsewa Amachitanso woona. A polygon otukukira kunja ndi amene ali:

• lililonse gawo kuti zikugwirizana ndi mfundo ziwiri mkati mwake, mabodza kwathunthu mu izo;

• m'menemo kugona diagonals zake zonse;

• iliyonse njingayo mkati saposa 180 °.

Polygon nthawi ukugawa ndege pawiri. Mmodzi wa iwo - okha (akhoza akachita bwalo), ndi zina - malire. Woyamba amatchedwa dera lamkati, ndipo chachiwiri - m'dera la kunja kwa chithunzi zojambula. Izi ndi mphambano ya polygon (mu mawu ena - mbali okwana) angapo theka ndege. Motero, aliyense gawo ndi malekezero pa mfundo zomwe a polygon kwathunthu chilichonse.

Zosiyanasiyana polygons otukukira kunja

Tanthauzo otukukira kunja polygon silimanena kuti pali mitundu yambiri ya iwo. Ndipo aliyense wa iwo ali ndi muyezo winawake. Choncho, polygons otukukira kunja, zimene ngodya mkati mwa 180 °, anatchula pang'ono otukukira kunja. The otukukira kunja zojambula chithunzi kuti aatali atatu, akutchedwa makona anayi - quadrilateral, zisanu - Pentagon, etc. Aliyense wa otukukira kunja N-gons likukwaniritsa zotsatirazi zofunika zofunika: .. N ayenera kukhala wolingana kapena kuposa 3. Aliyense wa Triangles ndi otukukira kunja. The zojambula chithunzi cha mtundu umene mfundo zonse zili pa bwalo, wotchedwa bwalo alembedwa. Anafotokoza otukukira kunja polygon amatchedwa ngati mbali zake zonse padziko bwalo kuti amugwire iye. polygons awiri amatchedwa ofanana Koma ngati pamene ntchito anakuta akhoza pamodzi. Lathyathyathya polygon otchedwa polygonal ndege (a ndege gawo) kuti zochepa geometrical chithunzi.

Zonse polygons otukukira kunja

polygons zonse wotchedwa akalumikidzidwa zojambula ndi kumathandiza kupeza ngodya zabwino ofanana ndi mbali. Mkati iwo pali mfundo 0, lomwe ndi mtunda wofanana ndi aliyense wa mfundo zake. Njirayi imatchedwa pakati pa chithunzi geometrical. Mizere wolumikiza malo ndi mfundo za chithunzi zojambula otchedwa apothem, ndi kulumikiza mfundo 0 ndi zipani - radii.

Kukhoza rectangle - lalikulu. Equilateral makona amatchedwa equilateral. Pakuti akalumikidzidwa ngati pali ulamuliro zotsatirazi: lililonse otukukira kunja polygon ngodya ndi 180 ° * (n-2) / N,

kumene N - chiwerengero cha mfundo za otukukira kunja zojambula chithunzi.

M'dera la polygon lililonse anatsimikiza mfundo:

S = tsa * H,

kumene tsa chofanana ndi ndalama theka mbali zonse polygon, ndi H ndi kutalika apothem.

Katundu otukukira kunja polygons

Otukukira kunja polygons ndi katundu wina. Choncho, gawo limene limalumikizana mfundo ziwiri za chithunzi zojambula, kwenikweni ili mmenemo. umboni:

Tiyerekeze kuti P - ndi otukukira kunja polygon. Tengani mfundo ziwiri lachabechabe, mwachitsanzo, A ndi B, amene ali a P. Mwa tanthauzo panopa ndi polygon otukukira kunja, mfundo zimenezi zili pa mbali imodzi ya mzere wolunjika kuti muli malangizo R. Choncho, AB alinso katundu uyu ndi ali R. A polygon otukukira kunja nthawi zonse akhoza kugawidwa mu Triangles angapo mwamtheradi diagonals onse, umene unachitikira wina wa mfundo zake.

Kumathandiza kupeza ngodya zabwino otukukira kunja akalumikidzidwa zojambula

The kumathandiza kupeza ngodya zabwino kwa polygon otukukira kunja - ndi kumathandiza kupeza ngodya zabwino zimene anapanga ndi zipani. ngodya mkati mwake muli m'dera mkati mwa chithunzi zojambula. Ngodya kuti aumbike mwa mbali zonse zomwe converge pa vertex, wotchedwa mbali ya polygon otukukira kunja. Ngodya pafupi ku ngodya mkati mwa chithunzi geometrical, wotchedwa kunja. Aliyense ngodya ya polygon otukukira kunja, anakonza mkati, ndi:

180 ° - ×

kumene × - phindu kunja ngodya. Izi chilinganizo yosavuta ndi ntchito mtundu uliwonse wa akalumikidzidwa zojambula zimenezi.

Ambiri, chifukwa ngodya kunja kulibe zotsatirazi ulamuliro: lililonse otukukira kunja polygon ngodya wofanana ndi kusiyana pakati pa 180 ° ndi mtengo wa mbali mkati. Zingabweretse mfundo kuyambira -180 ° mpaka 180 °. Choncho, pamene mbali wophiphiritsa 120 °, maonekedwe adzakhala ndi mtengo wa 60 °.

Uwerenge kumathandiza kupeza ngodya zabwino za polygons otukukira kunja

Uwerenge kumathandiza kupeza ngodya zabwino mkati mwa munthu polygon otukukira kunja udzakhazikitsidwe mfundo:

180 ° * (n-2),

kumene N - chiwerengero cha mfundo za N-gon.

Uwerenge kumathandiza kupeza ngodya zabwino kwa polygon otukukira kunja kuchita masamu ndithu chabe. Taganizirani aliyense ngati zojambula mawonekedwe. Kudziwa Uwerenge kumathandiza kupeza ngodya zabwino mu polygon otukukira kunja ayenera kugwirizana chimodzi mwa mfundo zake mfundo zina. Chifukwa cha zimenezi akutembenukira (n-2) ya makona lapansi. Amadziwika kuti Uwerenge kumathandiza kupeza ngodya zabwino za makona aliyense ali nthawizonse 180 °. Chifukwa chiwerengero chawo mu polygon aliyense wofanana (n-2), Uwerenge kumathandiza kupeza ngodya zabwino mkati mwa zimakwana liri 180 ° × (n-2).

Zopanda otukukira kunja polygon ngodya, ndicho aliyense zoyandikana mkati ndi kunja kumathandiza kupeza ngodya zabwino kwa iwo, mu izi otukukira kunja zojambula chithunzi nthawizonse kukhala wolingana 180 °. Pa mfundo imeneyi, tingadziwire Uwerenge m'makona ake onse:

180 × N.

Uwerenge kumathandiza kupeza ngodya zabwino mkati ndi 180 ° * (n-2). Choncho, Uwerenge ngodya zonse kunja kwa chithunzi anapereka njira:

180 ° * N-180 ° - (n-2) = 360 °.

Uwerenge kumathandiza kupeza ngodya zabwino kunja kwa polygon aliyense otukukira kunja nthawizonse kukhala wolingana 360 ° (kaya chiwerengero cha m'mbali mwake).

Kunja ngodya ya polygon otukukira kunja zambiri ukufanizidwa ndi kusiyana pakati ° 180 ndi mtengo wa mbali mkati.

katundu wina wa polygon otukukira kunja

Kuwonjezera pa katundu zofunika deta kanjedza zojambula, iwonso zina, zimachitika pamene muzizigwiritsa. Choncho, aliyense wa polygons akhoza unagawika angapo otukukira kunja N-gons. Kuti tichite zimenezi, kupitiriza aliyense wa m'mbali mwake ndi kudula mawonekedwe zojambula pankhani imeneyi molunjika. Unagawanika polygon aliyense mu magawo angapo otukukira kunja n'zotheka ndi kuti pamwamba pa aliyense wa ndalama lifanane ndi mfundo zake. Kuyambira chithunzi geometrical akhoza kukhala zosavuta kuti Triangles kudzera diagonals onse kuchokera vertex wina. Choncho, polygon aliyense pamapeto pake, akhoza kugawidwa mu nambala inayake ya Triangles, zomwe ndi zothandiza kuthetsa ntchito zosiyanasiyana zokhudzana akalumikidzidwa ngati geometrical.

Kukafika kutsogolo kwa polygon otukukira kunja

The zigawo za polyline ndi, polygon otchedwa zipani zambiri anasonyeza makalata zotsatirazi: AB, BC, ma cd, de, EA. tsidya lino la chithunzi geometrical ndi mfundo ndi, b, c, D, e. Uwerenge zotalika mbali ya polygon otukukira kunja amatchedwa wozungulira ake.

The circumference wa polygon ndi

Otukukira kunja polygons akhoza adalowa anafotokoza. Circle tangent m'mbali zonse za chithunzi zojambula, wotchedwa alembedwa mu izo. polygon iyi akutchedwa anafotokoza. Pakatikati pa bwalo lomwe linalembedwa mu polygon ndi mfundo za mphambano ya bisectors wa kumathandiza kupeza ngodya zabwino mkati anapatsidwa mawonekedwe zojambula. M'dera la polygon ndi wofanana:

S = tsa r Mukhoza *

kumene r Mukhoza - ndi utali wozungulira bwalo alembedwa, ndipo tsa - semiperimeter polygon izi.

bwalo A munali polygon mfundo, wotchedwa anafotokoza pafupi. Ndipotu, otukukira kunja zojambula chithunzi wotchedwa alembedwa. Pakatikati pa bwalo, amene akambidwa kuti zimenezi polygon ndi otchedwa mphambano mfundo midperpendiculars kumbali zonse.

Opendekera otukukira kunja akalumikidzidwa zojambula

The diagonals wa otukukira kunja polygon - mbali kuti zikugwirizana osati loyandikana mfundo. Aliyense wa iwo ali mkati chithunzi ichi zojambula. Chiwerengero cha diagonals wa N-gon wayikidwa malinga ndi njira:

N = N (n - 3) / 2.

chiwerengero cha diagonals wa polygon otukukira kunja amagwira ntchito yofunika kwambiri mu Masamu pulayimale. Chiwerengero cha Triangles (K), zomwe mwina kuswa iliyonse polygon otukukira kunja, masamu ndi chilinganizo zotsatirazi:

K = N - 2.

chiwerengero cha diagonals wa polygon otukukira kunja nthawi zonse amadalira chiwerengero cha mfundo.

Kugawa kwa polygon otukukira kunja

Nthawi zina, mavuto Masamu ntchito koyenera kuti athyole polygon otukukira kunja mu Triangles zingapo ndi diagonals sanali intersecting. Vutoli ingathetse kuchotsa chilinganizo ena.

KUFOTOKOZERA vuto: kuitana abwino kugawa kwa otukukira kunja N-gon mu Triangles angapo diagonals kuti intersect kokha pa mfundo za chithunzi zojambula.

Yankho: Tiyerekeze kuti P1, P2, P3, ..., Pn - pamwamba pa N-gon. Number Xn - chiwerengero cha partitions ake. Muziganizira chifukwa opendekera zojambula chithunzi ku Pihahiroti Pn. Mu aliyense wa partitions wokhazikika P1 Pn ali wa makamaka makona P1 ku Pihahiroti Pn, imene 1

Tiyeni ine = 2 gulu la partitions zonse, nthawi zonse munali opendekera P2 Pn. Chiwerengero cha partitions kuti zina mwa izo, wofanana ndi chiwerengero cha partitions (n-1) -gon P2 P3 P4 ... Pn. M'mawu ena, ndi wofanana Xn-1.

Ngati ine = 3, ndiye partitions zina gulu nthawizonse muli opendekera P3 P1 ndi P3 Pn. chiwerengero cha partitions zolondola kuti zili mu gulu, kodi lifanane ndi chiwerengero cha partitions (n-2) -gon P3, P4 ... Pn. M'mawu ena, zidzakhala Xn-2.

Lolani ine = 4, ndiye Triangles mwa kugawa zolondola mosakayikira muli kansalu P1 Pn P4, umene adjoin ndi quadrangle P1 P2 P3 P4, (n-3) -gon P5 P4 ... Pn. chiwerengero cha partitions zolondola quadrilateral ngati liri X4, ndipo chiwerengero cha partitions (n-3) -gon liri Xn-3. Kuchokera pa zimene takambiranazi, tikhoza kunena kuti chiwerengero cha partitions zonse zimene zili mu gulu ili liri Xn-3 X4. Kuyitana magulu ena, amene ine = 4, 5, 6, 7 ... amene muli 4 Xn-X5, Xn-5 X6, Xn-6 ... X7 partitions zonse.

Tiyeni ine = N-2, chiwerengero cha partitions zoona gulu watipatsa lifanane ndi chiwerengero cha partitions mu gulu, imene ine = 2 (mwa kulankhula kwina, liri Xn-1).

Popeza X1 = X2 = 0, X3 = 1 ndi X4 = 2, ..., chiwerengero cha partitions wa polygon otukukira kunja ndi:

Xn = Xn-1 + Xn-2 + Xn-3, Xn-X4 + X5 + 4 ... + X 5 + 4 Xn-Xn-X 4 + 3 + 2 Xn-Xn-1.

Mwachitsanzo:

X5 = X4 + X3 + X4 = 5

X6 = X4 + X5 + X4 + X5 = 14

X7 + X5 = X6 + X4 * X4 + X5 + X6 = 42

X7 = X8 + X6 + X4 * X5 + X4 * X5 + X6 + X7 = 132

Chiwerengero cha partitions zolondola intersecting mwa munthu opendekera

Pamene afufuze milandu munthu, zikhoza kapadera kuti chiwerengero cha diagonals wa otukukira kunja N-gon chofanana ndi mankhwala a partitions zonse chitsanzo chojambulidwa ichi (n-3).

Umboni wa kopanda izi: Tinene kuti P1n = Xn * (n-3), ndiye aliyense N-gon akhoza kugawidwa (n-2) kansalu. Pankhaniyi mmodzi wa iwo akhoza zakhala zikuzunza m'miyoyo (n-3) -chetyrehugolnik. Pa nthawi yomweyo, aliyense quadrangle ndi opendekera. Popeza ichi otukukira kunja zojambula chithunzi diagonals awiri angathe kuchitidwa, kutanthauza kuti aliyense (n-3) -chetyrehugolnikah angafunikire kuchititsa zina opendekera (n-3). Pa mfundo imeneyi, tikhoza kunena kuti pa kugawa lenileni ali ndi mwayi (n-3) -diagonali msonkhano zofunikira za ntchito imeneyi.

Area otukukira kunja polygons

Nthawi zambiri, pothetsa mavuto osiyanasiyana Masamu pulayimale pali kufunika kudziwa malo a polygon otukukira kunja. Musamachite (XI. Yi), ine = 1,2,3 ... N akuimira ndondomeko ya ndondomeko ya oyandikana nawo mfundo zonse polygon, wopanda kudzikonda mphambano. Pankhaniyi, malo ake ndi kuchita masamu ndi chilinganizo zotsatirazi:

_{S = ½ (Σ (X ine} + X ine + ₁₎ (Y _{ine +} Y _ine + _1)),

M'menemo (X _1, Y _{1) = (X N 1, Y} N + _1).