Yadda yin lissafi da yanki na wani dala: tushe, gefe da kuma full?

A shirye-shiryen da jarrabawa a ilmin lissafi dalibai dole systematize sanin aljabara da kuma lissafi. Ina son hada duk aka sani bayanai, kamar yadda za a lissafta da yanki na wani dala. Haka kuma, fara daga kasa da kuma gefen fuskantar har sai dukan surface area. Idan gefen fuskantar halin da ake ciki ya bayyana a sarari, kamar yadda su ne triangles, tushe ne ko da yaushe daban-daban.

Ta yaya ya zama lokacin da yankin na tushe na dala?

Yana iya zama quite wani adadi daga wani sabani alwatika ga n-gon. Kuma wannan tushe, fãce da bambanci a cikin yawan malã'iku, zai iya kasance daidai ko ba daidai ba adadi. A cikin sha'awa na dalibai ayyuka a kan jarrabawa sami kawai jobs da daidai Figures a gindi. Saboda haka, za mu kawai magana game da su.

equilateral alwatika

Wannan shi ne equilateral. Daya cewa duk jam'iyyun ne daidai da an tsara shi ta hanyar da harafin "a". A wannan yanayin, da tushe yanki na dala da aka lasafta ta da dabara:

S = (a ² * √3) / 4.

square

Da dabara ya lissafta ta yanki ne cikin sauki, shi ne "a" - gefe ne a sake:

Kuma S = ^2.

Hanawa yau da kullum n-gon

A gyaffan polygon guda nadi. Domin yawan kusassari amfani da Latin harafi n.

S = (n * a ²⁾ / (4 * KU (180º / n)) .

Yadda za a shiga a cikin lissafi na yankin na kaikaice da cikakken surface?

Tun da gindi adadi daidai ne, to, duk da fuskoki na dala ne daidai. Wanda kowannensu yana da wani isosceles alwatika, tun da gefen gefuna ne daidai. Sa'an nan, domin yin lissafi da yanki na wani gefe na dala bukatar dabara kunshi Naira Miliyan Xari monomials m. Yawan sharuddan da aka ƙaddara da adadin da tushe bangarorin.

A fannin wani isosceles Bamuda shi ne lissafta da dabara a cikin abin da rabi daga cikin tushe samfurin da aka halitta da tsawo. Wannan tsawo a cikin dala kira apothem. Its nadi - "A". A general dabara ga yankin na kaikaice surface ne kamar haka:

S = ½ P * A, inda P - kewaye da tushe daga cikin dala.

Akwai sau lokacin da shi ba a san su da tushe gefe, amma gefen gefuna ne (a) lebur da kwana a koli (α). Sa'an nan kuma ya dogara amfani da wadannan dabara yin lissafi da kaikaice yanki na dala:

S = n / 2 zuwa ² * zunubi α.

Task № 1

Yanayin. Nemo jimlar yankin na dala, idan karkashi ne wani equilateral alwatika tare da wani gefe na 4 cm kuma yana da darajar √3 apothem cm.

Rarrabẽwa. Ya kamata a fara da lissafi na da tushe kewaye. Tun da yake wannan ne na yau da kullum alwatika, sa'an nan P = 3 * 4 = 12 cm apothem Kamar yadda aka sani, wanda zai iya nan da nan lissafi da yankin na dukan kaikaice surface :. ½ * 12 * √3 = 6√3 ^cm2.

Don samun tushe alwatika ne da tamanin da area (4 ² * √3) / 4 = 4√3 ^cm2.

Domin sanin dukan yankin bukatar ninka biyu sakamakon dabi'u: 6√3 + 4√3 = 10√3 ^cm2.

Amsa. 10√3 ^cm2.

Matsala № 2

Yanayin. Akwai yau da kullum quadrangular dala. A tsawon tushe ne daidai da 7 mm, a kaikaice gefen - 16 mm. Kana bukatar ka sani ta surface area.

Rarrabẽwa. Tun da polyhedron - rectangular kuma daidai ne, a karkashi ne a square. Ji tushe yankin da kuma a kaikaice bangarorin ku iya lissafa cikin square dala. Da dabara ga square da aka ba a sama. Kuma na san duk gefen fuskoki na alwatika. Saboda haka, za ka iya amfani da Heron ta dabara domin kirga yankunan.

A farko lissafinta sauki da kuma kai wa ga wannan lambar: 49 mm ^2. Don lissafi na biyu darajar bukatar semiperimeter: (7 + 16 * 2): 2 = 19.5 mm. Yanzu za mu iya lissafta da yanki na wani isosceles alwatika: √ (19,5 * (19,5-7) * (19,5-16) ²⁾ = √2985,9375 = 54.644 mm ^2. Akwai hudu triangles, don haka a lokacin da kirga karshe lambobin za bukatar da za a yawaita da 4.

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

Amsa. 267,576 so darajar ² mm.

Task № 3

Yanayin. A yau da kullum quadrangular dala wajibi ne yin lissafi da yankin. An sani gefen square - 6 cm da tsawo - 4 cm.

Rarrabẽwa. A mafi sauki hanyar amfani da dabara zuwa samfurin na kewaye da apothem. A farko darajar da aka samu kawai. Na biyu kadan wuya.

Dole mu tuna da Pythagorean Theorem da la'akari da wani hakki ba alwatika. An kafa ta da tsawo daga cikin dala da kuma apothem, wanda shine hypotenuse. Na biyu kafa ne rabin gefen square, kamar yadda wani polyhedron tsawo da dama a tsakiyar shi.

Falala a kansu apothem (da hypotenuse wata dama alwatika) ne daidai √ (Maris ² + 4 ²⁾ = 5 (cm).

Yanzu yana yiwuwa yin lissafi da ake so darajar: ½ * (4 * 6) * 5 + 6 ² = 96 (cm ^2).

Amsa. 96 cm ^2.

Matsala № 4

Yanayin. Dana yau da kullum da kyakkyawan dala. A tarnaƙi daga karkashi daidaita zuwa 22 mm, a kaikaice gefuna - 61 mm. Mene ne yanki na kaikaice surface wannan polyhedron?

Rarrabẽwa. A tattaunawa a shi ne guda kamar yadda aka bayyana a cikin aiki №2. Kawai da dala da aka bai can zuwa square a tushe, da kuma a yanzu shi ne a heksagon.

A mataki na farko da aka lasafta ta da tushe yanki na sama dabara ^{(6 * 22 2) / (} 4 * KU (180º / 6)) = 726 / (tg30º) = 726√3 ^cm2.

Yanzu kana bukatar ka sami rabin-kewaye wani isosceles alwatika, wanda shi ne wani gefen fuska. (22 + 61 * 2) :. = 72 cm 2 zauna a Heron ta dabara yin lissafi da yankin na kowane daga cikin alwatika, sa'an nan ninka shi da shida Musulunci da kuma wanda ya juya ga tushe.

Lissafi a kan Heron ta dabara: √ (72 * (72-22) * ( 72-61) 2) = √435600 = 660 cm ^2. A lissafin da za su samar a kaikaice surface yankin: 660 * 6 = 3960 cm ^2. Ya zauna a ƙara su zuwa sami fitar da dukan surface: 5217,47≈5217 cm ^2.

Amsa. Filaye - 726√3 cm ^2, da gefen surface - 3960 cm ^2, da dukan yankin - 5217 cm ^2.