How kuverenga nzvimbo chepiramidhi: nechigadziko, divi uye azere?

Uchigadzirira bvunzo mu wemasvomhu vadzidzi vanofanira systematize ruzivo nemasvomhu uye geometry. Ndinoda chokubatanidza anozivikanwa mashoko ose, akadai sei kuverenga nzvimbo chepiramidhi. Uyezve, kubvira pasi uye divi anotarisana kusvikira pose nharaunda yose. Kana divi anotarisana nemamiriro ezvinhu chakajeka, sezvo ivo vari zvokuridza, nechigadziko anogara zvakasiyana.

Sei kuva apo nzvimbo mujinga chepiramidhi?

Zvinogona kuva zvikuru chipi nechipi kubva zvemasanga netriangle kuti N-gon. Uye nechigadziko ichi, kunze musiyano kwenhamba pamakona, zvingava zvakarurama kana kururama nhamba. In kufarira vadzidzi mabasa ari bvunzo kuwanikwa chete mabasa pamwe yakarurama manhamba pangoro. Saka, tichava chete kutaura nezvavo.

equilateral Triangle

Izvozvo equilateral. Mumwe kuti mapato ose vakaenzana uye anosarudzwa tsamba "munhu". Panyaya iyi, nechepasi nzvimbo chepiramidhi rinowanikwa ne nzira:

S = (a ² * √3) / 4.

sikweya

The yemaitiro kuverenga nzvimbo yaro iri nyore, iri "ari" - divi iri zvakare:

Uye S = ^2.

Zvemasanga dzose N-gon

At panhivi polygon chete izita. Nokuti uwandu pamakona vaishandisa Latin tsamba n.

S = (n * ²⁾ / (4 * nkm (180º / n)) .

How kupinda dzemaitirwe kwenzvimbo yacho lateral uye rizere pamusoro?

Sezvo mujinga nhamba wakarurama, ipapo zvose zviso chepiramidhi vakaenzana. Chimwe nechimwe chiri isosceles Triangle, kubva kurutivi micheto vakaenzana. Zvadaro, kuti ngaaverenge nzvimbo imwe divi chepiramidhi vanoda nzira zvinosanganisira mubatanidzwa monomials zvakafanana. Kuwanda okuti iri pakutsvaga chitsama pechingoro mumativi.

Nzvimbo imwe isosceles netriangle iri computed ne nzira iyo hafu zvitsigiro chigadzirwa zvikawanda ne kumusoro. Izvi nokukwirira piramidhi inonzi apothem. rokuti Its - "A". The vakawanda payakavakirwa nokuti nzvimbo lateral pamusoro riri sezvinotevera:

S = ½ P * A, apo P - mukomberedzo pakati mujinga chepiramidhi.

Pane nguva apo haisi kuzivikanwa kuti nechigadziko divi, asi kurutivi micheto vari (a) sandara, kundosvikazve pakona panguva chakaderedzwa mutengo (α). Zvadaro rinovimba kushandisa mutoo kuverenga lateral nzvimbo chepiramidhi:

S = n / 2 ² * chivi α.

Task № 1

Condition. Find zvachose nzvimbo chepiramidhi, kana chawo chiri chinhu equilateral netriangle ane divi 4 masendimita uye ane ukoshi √3 apothem cm.

Decision. Zvinofanira kutanga dzemaitirwe zvitsigiro mukomberedzo. Sezvo iri nguva, triangle, ipapo P = 3 * 4 = 12 CM apothem Sezvo chikazozikamwa, mumwe anogona pakarepo ngaaverenge nharaunda yose lateral pamusoro :. ½ * 12 * √3 = 6√3 ^cm2.

Kuti vawane nechepasi Triangle ndiye kukosha nzvimbo (4 ² * √3) / 4 = 4√3 ^cm2.

Kuti tizive nzvimbo yose vanofanira peta maviri zvichiguma tsika: 6√3 + 4√3 = 10√3 ^cm2.

Answer. 10√3 ^cm2.

Problem № 2

Condition. Pane quadrangular dzose chepiramidhi. Kureba pechingoro akaenzana 7 neMM, ari lateral pamupendero - 16 mm. Unofanira kuziva ayo pamusoro nharaunda.

Decision. Sezvo polyhedron - rectangular uye zvakarurama, panguva yayo iri sikweya. Kunzwa chawo nzvimbo uye lateral mativi kukwanisa kuverenga mativi piramidhi. The yemaitiro nokuti mativi kunopiwa kumusoro. Uye ndinoziva zvose parutivi zviso Triangle. Naizvozvo, unogona kushandisa Heron kuti nzira nokuda pakuverenga munzvimbo dzavo.

Masvomhu vokutanga ndivo nyore uye kungaita kuti nhamba iyi: 49 mm ^2. Kuverenga kukosha wechipiri vanoda semiperimeter: (7 + 16 * 2): 2 = 19.5 mm. Zvino tinogona ngaaverenge nzvimbo ane isosceles netriangle: √ (19,5 * (19,5-7) * (19,5-16) ²⁾ = √2985,9375 = 54.644 mm ^2. Kune zvokuridza zvina, saka kana pakuverenga kwokupedzisira nhamba vanofanira ngazviwande ne 4.

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

Answer. 267,576 akakumbira kukosha ² mm.

Task № 3

Condition. Nguva quadrangular Pyramid zvinokosha kuverenga nzvimbo. Zvinonzi anozivikanwa divi pachivara - 6 masendimita uye kukwirira - 4 cm.

Decision. Nyore nzira yokushandisa nzira kusvikira zvibereko mukomberedzo uye apothem. Chokutanga ukoshi inowanikwa nyore. Chechipiri zvishoma zvakaoma.

Tinotofanira kuyeuka Pythagorean theorem no uye funga netriangle zvakanaka. Zvinonzi inoumbwa kukwirira chepiramidhi uye apothem, rinova hypotenuse. The gumbo wechipiri hafu kurutivi square, sezvinoita polyhedron nokukwirira chikawira pakati payo.

Nyasha apothem (the hypotenuse pamusoro netriangle kurudyi) akaenzana √ (March ² + 4 ²⁾ = 5 (cm).

Zvino zvinoita ngaaverenge kutsvakwa ukoshi: ½ * (4 * 6) * 5 + 6 ² = 96 (masendimita ^2).

Answer. 96 masendimita ^2.

Problem № 4

Condition. Dana dzose hexagonal chepiramidhi. Mativi chawo akaenzana 22 neMM, ari lateral micheto - 61 mm. Chii nzvimbo lateral pose polyhedron ichi?

Decision. The pfungwa mairi vakafanana sezvinorondedzerwa basa №2. ari piramidhi chete ikapiwa yaiungana mujinga, uye zvino ari Hexagon.

Danho rokutanga rinowanikwa nevanhu muzasi nzvimbo pamusoro payakavakirwa ^{(6 * 22 2) / (} 4 * nkm (180º / 6)) = 726 / (tg30º) = 726√3 ^cm2.

Zvino unofanira kuwana hafu-mukomberedzo ane isosceles Triangle, ndiro divi chiso. (22 + 61 * 2) :. = 72 masendimita 2 kunogara pamusoro Heron kuti bumbiro remitemo ngaaverenge nzvimbo nerimwe Triangle, chipeta parutivi nedanga matanhatu uye uya akatendeuka akabuda chawo.

Masvomhu musi Heron kuti nzira: √ (72 * (72-22) * ( 72-61) 2) = √435600 = 660 masendimita ^2. Masvomhu kuti ichapa lateral pamusoro nharaunda: 660 * 6 = 3960 masendimita ^2. Zvazvinosarira kuwedzera akavakumikidza kuti uzive rose pamusoro: 5217,47≈5217 masendimita ^2.

Answer. Grounds - 726√3 masendimita ^2, kudivi pamusoro - 3960 masendimita ^2, nharaunda yose - 5217 masendimita ^2.