Pehea e huli i ka wahi o ka quadrilateral?

Inā ka pelane, ua kūpono unuhi mau Hoʻohana no laila i kekahi e hoʻomaka ma ka wahi, kahi o ka mua kekahi i pau ai, ua loaa kekahi lahuikanaka i mokumokuia laina. Mau Hoʻohana i kapaʻia nā loulou, a me nā wahi ma kahi e intersect - mau poʻo. A hiki i ka hopena o ka hope Hoʻohana intersects ka mua ho omaka, ua loaa i kekahi paniʻia laina, a i hookaawale ai i ka pelane, i loko o nā māhele. Kekahi o ia mea finite, a me ka lua o ka mana loa.

Simple paʻa hālawa me ka puni iho la ka hapa o ka pelane (ia i mea finite) ua kapaia he iieeaii. Nā Hoʻohana mau aoao elua, a me ka pana pua i hanaʻia e ia - mau poʻo. I ka helu o ka aoao o kekahi iieeaii like me ka helu ana o vertices. A huahelu a he ekolu aoao, i kapaʻia he triangle, akā, eha - he quadrilateral. Iieeaii heluna hapauuku wehewehe aku ma ka mea nui, e like me ka wahi i hōʻike i ka nui o ka huahelu. Pehea e huli i ka wahi o ka quadrilateral? Ao mai la ia ma kekahi lala o ka makemakika - anahonua.

E huli i ka wahi o ka quadrilateral, ia mea e pono ke ike i ka 'ano mea a pau - convex paha nonconvex? Convex iieeaii a pau, kŘpa a pololei (a me ka mea pono no i kekahi o na aoao elua) ma kaʻaoʻao hookahi. Eia kekahi, eia ke ano o ka quadrilaterals like me ka parallelogram me ka ae like a laua e ku pono ana aoao (ʻano iā ia rectangle me ka pololei kihi, rhombus me ka like aoao, heʻahā like me nā mea a pau pana pua'ākau, a me ka eha like aoao), trapezoid me nā laua e ku pono anaʻaoʻaoʻelua, a deltoid me nā hui o ka pili aoao i like.

Mika pāhoʻonui lua kekahi iieeaii e hoʻohana i ka hoʻomaopopo like iaoia, i mea e uhai ia i loko o triangle mauna, kēlā triangle ho omaulia ākeʻakeʻa kumu wahi a pa i kēia mau hopena. Kekahi convex quadrilateral ua maheleia i mau triangle mauna, nonconvex - elua a ekolu o ka triangle, i ka wahi o ia i loko o keia hihia e komo i ka huina a me ka koena unuhi o ka hopena. Ke kahi ana o kekahi triangle he pōpilikia me ka hapalua o ka waihona ipu huahana o (i) ke kiekie (H), lawe mai ana i ka waihona ipu. Ka haʻilula i ua hoʻohana 'ia ma kēia hihia no ka mea i ka ho omaulia ua palapala like: S = ½ • ka • H.

Pehea e huli i ka wahi o ka quadrilateral, no kekahi laʻana, he parallelogram? He mea pono e ike i ka loa o ka waihona ipu (i), heʻaoʻao lōʻihi (ƀ) a loaʻa i ka sine o ka huina α, i hanaʻia ma ke kumu, a me kaʻaoʻao (sinα), no ka helu ana i ka haʻilula mea like: S = ka • ƀ • sinα. Mai ka sine o ka huina α o ka huina hoonui o kekahi kumu o ka parallelogram ma luna o kona kiekie (H = ƀ) - he laina p ÷ haku i ka waihona ipu, kona wahi he pōpilikia e hoonui i ke kiʻekiʻe o kona kumu: S = ka • H. No ka ho omaulia i ka wahi o ka rhombus, a me ka rectangle i fits keia haʻilula. Mai ke kīloi paʻewaʻaoʻao o ka rectangle coincides me ke kiʻekiʻe ƀ H, kona wahi he pōpilikia ma ka haʻilula S = ka • ƀ. Ka wahi o ka huinahalike, no ka mea, he = ƀ, e like me ka hale o konaʻaoʻao: S = i • i = a² . Ka wahi o ka trapezoid he pōpilikia me ka hapalua o ka huina o kona aoao, hoonuiia ma ke kiekie, (ka mea, ua mālama 'ia ai i ke kumu o ka trapezoid ÷ haku i): S = ½ • (he + ƀ) • H.

Pehea e huli i ka wahi o ka quadrangle, ina i ike ole loa o kona aoao, akā, ua ikeia no kona diagonal (e) a me (f), a me ka sine o ka huina α? Ma keia hihia i ka wahi he pōpilikia me ka hapalua o ka huina hoonui o kona diagonals (ka laina e hoʻohui i nā vertices o ka iieeaii), hoonuiia e ka sine o ka huina α. Ke haʻilula hiki ke kākau i loko o kēia palapala: S = ½ • (e • f) • sinα. I mau rhombus āpana ma keia hihia, e like no ka hapalua o ka huina hoonui o ka diagonals (na laina ai pili aku ku pono kihi o ka rhombus): S = ½ • (e • f).

Pehea e huli i ka wahi o ka quadrilateral, i mea ole he parallelogram a me ka trapezoid, olelo ka mea, ua haawiia i like ka ākeʻakeʻa kumu rectangle. Ka wahi o ka huahelu i hoike i ka olelo o kona hapa-anapuni (Ρ - i ka huina o nāʻaoʻaoʻelua me ka hoʻomaopopo like vertex), i ka aoao i kekahi, ƀ, pela aku, n, a me ka huina o elua e ku pono pana pua (α + β): S = √ [(Ρ - he) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d) - i • ƀ • pela aku • b • cos² ½ (α + β)].

Inā quadrilateral kākauʻia i loko o ka poai nei, a φ = 180 °, ma ka mea e hoʻomaulia i kona wahi hoʻohana Brahmagupta haʻilula (Mauna astronomer a me ka mathematician, ka poe i noho ma 6-7 kenekulia AD): S = √ [(Ρ - he) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d)]. Inā quadrilateral hōʻike anapuni, laila, (i + e = ƀ + d), a me kona wahi he pōpilikia: S = √ [he • ƀ • c. • D] • hewa ½ (α + β). Inā ka quadrangle e ka poʻe i ho'ākāka 'ia i kekahi hoa kanaka, a me nā kākauʻia pōʻai i ka' ē aʻe, me ka wahi hoʻohana e hoʻomaulia i ka kēia haʻilula: S = √ [he • ƀ • pela aku • b].