Kodi kuwerengera malo a piramidi: m'munsi, mbali ndi zonse?

Pokonzekera mayeso ophunzira masamu kuti dongosolo chidziwitso cha ajebura ndi masamu. Ndikufuna kuphatikiza mfundo zonse zodziwika, monga mmene kuwerengera malo a piramidi. Komanso, kuyambira pansi ndi mbali loyang'anizana mpaka m'dera lonse pamwamba. Ngati mbali nkhope zinthu bwino, ngati iwo ali Triangles, m'munsi ndi nthawi zosiyana.

Kodi kukhala m'dera la m'munsi mwa piramidi?

Zimakhala ndithu chithunzi chilichonse ku chinthu cha makona umasinthasintha ku N-gon. Ndi tsinde izi, kupatula kusiyana kwa chiwerengero cha kumathandiza kupeza ngodya zabwino, angakhale olondola kapena pachithunzichi chithunzi. Mu chidwi cha ntchito ophunzira mayeso anapeza ntchito yekha ndi kanjedza zoona m'munsi lapansi. Choncho, tidzakhala kulankhula za iwo.

equilateral makona

Kuti ndi equilateral. Amene maphwando onse ndi ofanana ndipo zinaperekedwa mwa kalata "a". Pankhaniyi, malo m'munsi mwa piramidi kuchita masamu mfundo:

S = (a ² * √3) / 4.

khwalala

Ndondomeko kuwerengera malo ake ndi losavuta, ndi "a" - mbali kachiwiri:

Ndipo S = ^2.

Umasinthasintha zonse N-gon

Pa m'mbali mwa polygon akuti chomwecho. Chiwerengero cha kumathandiza kupeza ngodya ntchito Latin kalata N.

S = (n * ²⁾ / (4 * tg (180º / n)) .

Kodi kulowa mu mawerengedwe a m'dera la lateral ndi zonse padziko?

Popeza m'munsi zili zolondola, ndiye nkhope yonse ya piramidi ofanana. Omwe ndi isosceles makona atatu, kuyambira mbali m'mbali ofanana. Ndiye, kuti kuwerengetsa dera mbali ya piramidi ayenera chilinganizo wopangidwa Muwerenge monomials zofanana. chiwerengero cha mawu anatsimikiza ndi kuchuluka kwa mbali m'munsi.

Dera ndi makona isosceles ndi yopangidwa ndi njira imene theka la mankhwala m'munsi ndi kuchulukitsa ndi kumwamba. kutalika izi mu piramidi otchedwa apothem. chikhazikitso ake - "A". Chilinganizo ambiri dera la pamwamba lateral chiri motere:

S = ½ P * A, kumene P - kukafika kutsogolo kwa m'munsi mwa piramidi.

Pali nthawi zina pamene si kudziwika kwa mbali m'munsi, koma mbali m'mbali ndi (a) mosabisa komanso mbali pa pamwamba (α). Ndiye iwo amadalira ntchito mfundo zotsatirazi kuwerengera m'dera lateral wa piramidi:

S = N / 2 ² machimo α.

Ntchito № 1

Chikhalidwe. Pezani malo okwana piramidi, ngati maziko ake ali ndi equilateral makona ndi mbali ya masentimita 4 ndipo ali mtengo √3 apothem cm.

Kusankha. Iwo ayenera kuyamba ndi mawerengedwe a kukafika kutsogolo m'munsi. Popeza ndi kansalu zonse, ndiye P = 3 4 = 12 masentimita apothem Monga amadziwika, wina athanso kuwerengera dera lonse lateral pamwamba :. ½ * 12 * √3 = 6√3 ^cm2.

Kuti mupeze m'munsi makona ndi mtengo wa m'deralo (4 ² * √3) / 4 = 4√3 ^cm2.

Kudziwa m'dera lonse ayenera pindani awiri chifukwa makhalidwe: 6√3 + 4√3 = 10√3 ^cm2.

Yankho. 10√3 ^cm2.

Vuto № 2

Chikhalidwe. Pali nthawi zonse quadrangular piramidi. Kutalika kwa m'munsi ndi wofanana ndi 7 mm, m'mphepete lateral - 16 mm. Muyenera kudziwa padziko dera lake.

Kusankha. Popeza polyhedron - amakona anayi ndi zolondola, pa maziko ake ndi lalikulu. Kumva m'munsi m'dera ndi mbali lateral athe kuwerenga piramidi lalikulu. Tikutsata ndondomeko za lalikulu zapatsidwa pamwambazi. Ndipo ine ndikudziwa nkhope onse mbali ya makona lapansi. Chotero, inu mukhoza kugwiritsa ntchito chilinganizo chimeza kwa kuwerengetsa madera awo.

Kuwerengetsera woyamba osavuta ndi kuyambitsa nambala iyi: 49 mamilimita ^2. Kuwerengetsa phindu lachiwiri ayenera semiperimeter: (7 + 16 * 2): 2 = 19.5 mm. Tsopano ife tikhoza kuwerengera m'dera la makona isosceles: √ (19,5 * (19,5-7) * (19,5-16) ²⁾ = √2985,9375 = 54.644 mamilimita ^2. Pali Triangles anayi, pamene kuwerengetsa manambala chomaliza adzafunika kuchulukitsa ndi 4.

Analandira: 49 + 4 * 54,644 = 267,576 ^mm2.

Yankho. 267,576 anakhumba kufunika ² mm.

Ntchito № 3

Chikhalidwe. Pa zonse quadrangular piramidi m'pofunika kuwerengetsa m'deralo. Amadziwika mbali ya lalikulu - masentimita 6 ndi kutalika - 4 cm.

Kusankha. Chophweka njira kugwiritsa ntchito njira za mankhwala a kukafika kutsogolo ndi apothem. Mtengo Woyamba umapezeka chabe. The chachiwiri pang'ono kwambiri.

Ife kukumbukira Lingaliro Lovomerezeka Pythagorean ndi kuganizira kansalu bwino. Iwo aumbike mwa kutalika kwa piramidi ndi apothem, ndilo hypotenuse. The mwendo wachiwiri ndi mbali theka a lalikulu, ngati polyhedron kutalika imagwera pakati pa izo.

Woyanjidwa apothem (a hypotenuse wa kansalu kumanja) ndi wofanana √ (3 ² + 4 ²⁾ = 5 (cm).

Tsopano n'zotheka kuwerengera phindu amafuna: ½ * (4 * 6) * 5 + 6 ² = 96 (masentimita ^2).

Yankho. 96 masentimita ^2.

Vuto № 4

Chikhalidwe. Dana zonse hexagonal piramidi. Mbali ya maziko ake wolingana 22 mm, m'mphepete lateral - 61 mm. ndi m'dera la pamwamba lateral wa polyhedron chiyani?

Kusankha. The kuganiza kuti ali yemweyo monga afotokozera mu ntchito №2. Only piramidi anapatsidwa kumeneko lalikulu pa maziko, ndipo tsopano ndi hexagon.

Chinthu choyamba kuchita masamu ndi dera m'munsi mwa njira pamwamba ^{(6 * 22 2) / (} 4 * tg (180º / 6)) = 726 / (tg30º) = 726√3 ^cm2.

Tsopano inu muyenera kupeza theka kukafika kutsogolo kwa chinthu cha makona isosceles, lomwe ndi mbali nkhope. (22 + 61 * 2) :. = 72 masentimita 2 ukhala pa chilinganizo chimeza ndi kuwerengera m'dera lililonse la makona, ndiyeno kuchulukitsa izo ndi khola isanu ndi umodzi kuti kunapezeka kuti maziko a.

Asayansi pa chilinganizo chimeza akuti: √ (72 * (72-22) * ( 72-61) 2) = √435600 = 660 masentimita ^2. kuwerengetsera kuti adzapereka lateral padziko dera: 660 * 6 = 3960 masentimita ^2. Mpaka kuwonjezera iwo kupeza padziko lonse: 5217,47≈5217 masentimita ^2.

Yankho. Grounds - 726√3 masentimita ^2, mbali pamwamba - 3960 masentimita ^2, dera lonse - 5217 masentimita ^2.