I-polygon ejwayelekile. Inani lezinhlangothi zepoloni evamile

I-Triangle, isikwele, iheksagoni - lezi zibalo ziyaziwa cishe wonke umuntu. Kodwa akubona wonke umuntu owaziyo ngokuthi i-polygon ejwayelekile ingakanani. Kodwa lawa onke amanani afanayo wejometri. I-polygon evamile yinye enezingalo nezinhlangothi ezilinganayo. Kunezinombolo eziningi ezinjalo, kodwa zonke zinempahla efanayo, futhi amafomu afanayo afaka kubo.

Impahla yama-polygons avamile

Noma iyiphi i-polygon ejwayelekile, kungaba isikwele noma i-octagon, ingabhalwa embuthanweni. Le ndawo eyisisekelo ivame ukusetshenziswa lapho ukwakha umumo. Ngaphezu kwalokho, umbuthano ungabhalwa ne-polygon. Kulesi simo, inani lamaphoyinti oxhumana nabo lizolingana nenani lamacala awo. Kubalulekile ukuthi umbuthano obhalwe phansi kwi-polygon kuyoba nendawo ejwayelekile nayo. Lezi zibalo zejometri zixhomeke ku-oneorem eyodwa. Noma yikuphi uhlangothi lwe-n-gon ejwayelekile lixhunywe ne-radius ye-osokile R ejikelezwe ngakho ngakho-ke, ingabalwa ngokusebenzisa ifomula elandelayo: a = 2R ∙ sin180 °. Ngaphakathi kwendilinga yombuthano, awukwazi ukuthola izinhlangothi kuphela, kodwa futhi nomjikelezo we-polygon.

Indlela yokuthola inombolo yezinhlangothi ze-polygon ejwayelekile

Noma iyiphi i- n-gon ejwayelekile iqukethe izinombolo ezilinganayo ezijoyina ndawonye zakha umugqa ovaliwe. Kulokhu, wonke ama-angles omfanekiso obumbwe anenani elifanayo. Ama-Polygons ahlukaniswe abe alula futhi anzima. Iqembu lokuqala lihlanganisa unxantathu nesigcawu. Amaphonikoni anamanzi anamahlangothi amaningi. Zibandakanya nezinombolo ze-stellate. Kwama-polygoni avamile, izinhlangothi zitholakala ngokuzifaka embuthanweni. Sinikeza ubufakazi. Dweba i-polygon ejwayelekile ngezinhlangothi ezingenakubalwa n. Chaza umbuthano owuzungezayo. Cacisa irejista R. Manje cabanga ukuthi enye i-n-gon inikezwa. Uma amaphuzu angama-angles ayolala embuthanweni futhi alingana nomunye nomunye, khona-ke izinhlangothi zingatholakala ngefomula: a = 2R ∙ sinα: 2.

Ukuthola inani lamacala enxantathu elungile elibhalwe phansi

Unxantathu olinganayo uyingqamuzana evamile. Amafomu okusebenza kuwo asebenza okufana nesikwele, n-gon. Unxantathu uzobhekwa njengelungile uma unebude obufanayo eceleni. Ama-angles alingana no-60. Sakha unxantathu ngezinhlangothi zobude obunikeziwe a. Ukwazi ukulinganisa nokuphakama kwawo, umuntu angathola ukubaluleka kwezinhlangothi zawo. Ukuze senze lokhu, sisebenzisa indlela yokuthola ngokusebenzisa ifomu a = x: cosa, lapho x kungumphakathi noma ukuphakama. Njengoba zonke izinhlangothi zendxantathu zilingana, sithola = = b = c. Khona-ke ukuqinisekiswa okulandelayo kuzobamba: a = b = c = x: cosa. Ngokufanayo, umuntu angathola ukubaluleka kwezinhlangothi ezinxantathu ze-isosceles, kodwa x kuzoba ukuphakama okunikeziwe. Kulesi simo, kufanele kufakwe ngokucacile phansi kwesibalo. Ngakho, ukwazi ukuphakama x, sithola ohlangothini lwezinxantathu ze-isosceles ngefomula a = b = x: cosa. Ngemuva kokuthola ukubaluleka kwe-a, singakwazi ukubala ubude besisekelo c. Sisebenzisa i-theorem yasePythagoras. Sizofuna ukubaluleka kwesigamu sesisekelo c: 2 = √ (x: cosa) ^ 2 - (x ^ 2) = √x ^ 2 (1 - cos ^ 2a): cos ^ 2α = x ∙ tgα. Bese c = 2xtgα. Ngale ndlela elula umuntu angathola inombolo yamacala enoma iyiphi i-polygon eqoshiwe.

Ibala izinhlangothi zesiteji esibhalwe kumbuthano

Njenganoma iyiphi enye i-polygon eqoshiwe, isikwele sinamacala angalingana nama-angles. Amafomu afanayo afaka kuwo njengokungathi kunxantathu. Bala izinhlangothi zesigcawu kungaba ngesilinganiso se-diagonal. Ake sicabangele le ndlela ngokuningiliziwe. Kuyaziwa ukuthi i-diagonal ihlukanisa i-angle ngesigamu. Ekuqaleni, inani lalo laliyizi-90 degrees. Ngakho, ngemva kokuhlukaniswa, kwakhiwa ama-triangles amabili angama-rectangular. Amakhonksi abo e-base azolingana nama-45 degrees. Ngakho-ke, uhlangothi ngalunye lwesigcawu luzolingana, okungukuthi: a = c = c = q = e ∙ cosa = e√2: 2, lapho u-diagonal wesigcawu, noma isisekelo sexantathu elungile esakhiwe ngemuva kokuhlukaniswa. Lena akuyona yindlela kuphela yokuthola izinhlangothi zesigcawu. Sizobhala lesi sibalo kumbuthano. Ukwazi i-radius yalo mbuthano R, sithola uhlangothi lwesigcawu. Sizolibala ngale ndlela elandelayo: a4 = R√2. I-radii yama-polygons avamile ibalwa ngefomula R = a: 2tg (360 ^o: 2n), lapho ubude obude khona.

Indlela yokubala umjikelezo we-n-gon

Umjikelezo we-n-gon uyisibalo sezinhlangothi zawo zonke. Bala ukuthi akunzima. Ukwenza lokhu, udinga ukwazi ukuthi zonke izinhlangano zisho ukuthini. Kwezinhlobo ezithile zama-polygons, kukhona amafomula akhethekile. Bakuvumela ukuba uthole ngokushesha umjikelezo. Kuyaziwa ukuthi i-polygon ejwayelekile inezinhlangothi ezilinganayo. Ngakho-ke, ukuze uhlole umjikelezo wayo, kwanele ukwazi okungenani oyedwa wabo. Ifomula izoxhomeka enani lezinhlangothi zesibalo. Ngokuvamile, kubonakala kanje: P = an, lapho kukhona khona ukulinganisa, futhi n inombolo yama-angles. Isibonelo, ukuthola i-octagon ejwayelekile ehlangothini lwe-3 cm, uwandise ngo-8, okungukuthi, P = 3 ∙ 8 = 24 cm. Iheksagoni enehlangothini lika-5 cm, ubale: P = 5 ∙ 6 = 30 cm. I-polygon ngayinye.

Ukuthola umjikelezo we-parallelogram, isikwele, ne-rhombus

Kuye ukuthi zingaki izinhlangothi i-polygon ejwayelekile enezibalo, funda umjikelezo wayo. Lokhu kwenza lula umsebenzi. Ngempela, ngokungafani nezinye izibalo, kulokhu awudingi ukubuka zonke izinhlangothi zawo, eyodwa kuphela. Ngomgomo ofanayo, sithola umjikelezo we-quadrangles, okungukuthi, isikwele ne-rhombus. Naphezu kokuthi lezi zibalo ezifani, i-formula yabo i-P = 4a, lapho kukhona ohlangothini. Ake sinike isibonelo. Uma uhlangothi lwedayimane noma isikwele lingama-6 cm, bese sithola umjikelezo ngale ndlela elandelayo: P = 4 ∙ 6 = 24 cm. Ku-parallelogram, izinhlangothi ezihlukene kuphela zilingana. Ngakho-ke, umjikelezo wayo utholakala usebenzisa indlela ehlukile. Ngakho-ke, sidinga ukwazi ubude be-a nobubanzi besibalo. Khona-ke sisebenzisa ifomula P = (a + b) ∙ 2. I-parallelogram, lapho izinhlangothi zonke nezingalo zilingana, kuthiwa i-rhombus.

Ukuthola umjikelezo wezinxantathu ezilinganayo kanye noxantathu olungakwesokudla

I-perimeter yengxenyana ejwayelekile yokulinganisa ingatholakala ngefomula P = 3a, lapho kukhona ubude obude. Uma engaziwa, itholakala nge-median. Enxantathu yangemuva, izinhlangothi ezimbili kuphela zinenani elinganayo. Isisekelo sitholakala nge-Pythagorean theorem. Ngemuva kokuthi amanani azo zonke izinhlangothi ezintathu aziwa, balala umjikelezo. Kungatholakala ngokusebenzisa ifomu P = a + b + c, lapho i-a ne b ibalwa khona, futhi c isisekelo. Khumbula ukuthi kunxantathu ye-isosceles a = b = a, bese u-+ b = 2a, bese u-P = 2a + c. Isibonelo, ohlangothini lwesigxathu se-isosceles singu-4 cm, sithola isisekelo saso kanye nesigamu ngasinye. Sibala inani le-hypotenuse ngokusho kwethempythi ye-Pythagorean nge c = √a ² + ku- ² = √16 + 16 = √32 = 5.65 cm Manje faka umjikelezo P = 2 ∙ 4 + 5.65 = 13.65 cm.

Indlela yokuthola izingxenyana ze-polygon ejwayelekile

I-polygon ejwayelekile ikhona empilweni yethu nsuku zonke, isibonelo, isikwele esijwayelekile, unxantathu, i-octagon. Kubonakala sengathi akukho okulula kunokwakha lesi sibalo ngokwakho. Kodwa nje nje ekuboneni kokuqala. Ukuze wakhe noma yikuphi i-n-gon, kubalulekile ukwazi inani lezingalo zalo. Kodwa ungayithola kanjani? Ngisho nososayensi basendulo bazama ukwakha ama-polygoni avamile. Baqagela ukuwavumelanisa nabo embuthanweni. Bese babhala amaphuzu adingekayo kuwo, bawaxhuma ngemigqa eqondile. Ngezibalo ezilula, inkinga yokwakha yaxazululwa. Amafomula nama-theorems atholakala. Isibonelo, u-Euclid emsebenzini wakhe odumile othi "The Beginning" wayehlanganyele ekuxazululeni izinkinga ze-3-, 4-, 5-, 6- no-15-gons. Wathola izindlela zokwakha nokuthola ama-angles. Cabanga ukuthi ungakwenza kanjani lokhu nge-15-gon. Okokuqala udinga ukubala inani lama-angles angaphakathi. Kubalulekile ukusebenzisa ifomu S = 180⁰ (n-2). Ngakho-ke, sinikezwa i-15-gon, ngakho inombolo engu-15. Sifaka esikhundleni se-data esiyaziwa kwifomu bese sithola i- S = 180⁰ (15 - 2) = 180⁰ х 13 = 2340⁰. Sithole inani lazo zonke izingalo zangaphakathi ze-15-gon. Manje udinga ukuthola inani ngalinye. Ama-total angles 15. Ingabe ukubalwa kwe-2340⁰: 15 = 156⁰. Ngakho-ke, i-engeli ngayinye yangaphakathi ngu-156⁰, manje ngosizo lomlawuli nekhampasi ungakha i-15-gon efanele. Kodwa kuthiwani nge-n-gons eziyinkimbinkimbi? Kwaphela amakhulu eminyaka ososayensi baye bazama ukuxazulula le nkinga. Itholakala kuphela ngekhulu le-18 nguCarl Friedrich Gauss. Wakwazi ukwakha i-65537-gon. Kusukela ngaleso sikhathi, inkinga ibhekwa ngokusemthethweni ngokuxazululwa ngokugcwele.

Ukubalwa kwama-angles we-n-gons kuma-radians

Yiqiniso, kunezindlela eziningana zokuthola ama-polygons. Ngokuvamile zibalwa ngamadigri. Kodwa ungawaveza kuma-radians. Ungakwenza kanjani lokhu? Kubalulekile ukuqhubeka kanje. Okokuqala, sithola inani lezinhlangothi ze-polygon njalo, bese sisusa kuso 2. Ngakho, sithola inani: n - 2. Hlanganisa umehluko nge n ("pi" = 3.14). Manje kusele kuphela ukuhlukanisa umkhiqizo otholiwe ngenani lama-angles ku-n-gon. Cabangela lezi zibalo ngesibonelo sezinxantathu ezinamacala ayishumi nanhlanu. Ngakho-ke inombolo inombolo engu-15. Masisebenzise ifomu S = n (n - 2): n = 3,14 (15 - 2): 15 = 3,14 ∙ 13: 15 = 2,72. Yiqiniso, lokhu akuyona indlela kuphela yokubala i-angle kuma-radians. Ungamane uhlukanise ubukhulu be-engeli ngamadijithi ngenombolo 57.3. Phela, ama-degree amaningi kangaka afana ne-radon eyodwa.

Ukubalwa kwama-angles emahlathini

Ngaphezu kwamadigri nama-radians, ungazama ukuthola ama-angles we-polygon evamile ngesichotho. Lokhu kwenziwa kanje. Kusukela inani eliphelele lama-angles, khipha 2, uhlukanise umehluko ophumayo ngenani lamacala e-polygon ejwayelekile. Umphumela wanda ngamakhulu angu-200. Ngendlela, leyo unit of measurement of angles, njengesichotho, ayifuni ukusetshenziswa.

Ukubalwa kwama-angles angaphandle we-n-gons

Noma iyiphi i-polygon ejwayelekile, ngaphandle kweyodwa, kungenzeka ukubala futhi i-angi yangaphandle. Incazelo yalo itholakala ngendlela efanayo nayo yonke le mibalo. Ngakho-ke, ukuthola ikona yangaphandle ye-polygon ejwayelekile, udinga ukwazi incazelo ye-polygon yangaphakathi. Ngaphezu kwalokho, siyazi ukuthi inani lalezi zingalo ezimbili lihlala liyi-180 degrees. Ngakho-ke, senza izibalo ngale ndlela: 180⁰ inciphisa inani le-angi yangaphakathi. Sithola umehluko. Kuzolingana nenani le-angle elikude nayo. Isibonelo, ingaphakathi langaphakathi lesikwele ngamagremu angu-90, khona-ke ikona langaphandle lizoba 180⁰-90⁰ = 90⁰. Njengoba sibona, akunzima ukukuthola. I-angi yangaphandle ingathatha inani kusuka ku- + 180⁰ ukuya, ngokulandelana, -180 .