Olee gbakọọ ebe a pyramid: isi, n'akụkụ na zuru?

Na nkwadebe maka udomo oro na mgbakọ na mwepụ na ụmụ akwụkwọ ga-eme systematize ihe ọmụma nke algebra na jiometrị. Ga-amasị m ikpokọta niile mara na ozi, dị ka otú gbakọọ ebe a pyramid. Ọzọkwa, malite na ala na akụkụ chere ihu ruo mgbe dum n'elu ebe. Ọ bụrụ n'akụkụ chere ihu na ọnọdụ doro anya, dị ka ha na-triangles, ndị isi bụ mgbe dị iche iche.

Olee mgbe mpaghara nke isi nke pyramid?

Ọ pụrụ ịbụ nnọọ ihe ọ bụla na ọnụ ọgụgụ site na aka ike triangle na n-gon. Na nke a isi, ma e wezụga ihe dị iche na ọnụ ọgụgụ nke akụkụ, nwere ike ịbụ ziri ezi ma ọ bụ na-ekwesịghị ịdị na ọnụ ọgụgụ. Na mmasị nke ụmụ akwụkwọ ihe aga-eme na udomo oro hụrụ naanị ọrụ na ezi ọgụgụ na isi. Ya mere, anyị ga na-ekwu okwu banyere ha.

equilateral triangle

Nke ahụ bụ equilateral. One na niile ọzọ hà ma na-kpọrọ akwụkwọ ozi "a". Na nke a, ndị isi nke ebe pyramid na gbakọọ site usoro:

S = (a ² * √3) / 4.

square

The usoro gbakọọ ya ebe bụ mfe, bụ "a" - n'akụkụ bụ ọzọ:

Na S = ^2.

Aka ike mgbe nile n-gon

Ke n'akụkụ nke polygon otu aha. N'ihi na ọnụ ọgụgụ nke akụkụ-eji Latin akwụkwọ ozi n.

S = (n * a ²⁾ / (4 * tg (180º / n)) .

Olee otú ịbanye na ngụkọta oge nke ebe mpụta na zuru n'elu?

Ebe ọ bụ na ndị isi ọgụgụ ziri ezi, mgbe ahụ niile ihu nke pyramid hà. Onye ọ bụla n'ime nke bụ ihe isosceles triangle, ebe n'akụkụ n'ọnụ hà. Mgbe ahụ, iji gbakọọ ebe nke a n'akụkụ nke pyramid mkpa usoro esịnede nchikota nke monomials yiri. The ọnụ ọgụgụ nke okwu kpebisiri ike site ego nke isi n'akụkụ.

The ebe nke ihe isosceles triangle na-agbakọrọ site usoro nke ọkara nke isi ngwaahịa na-uba site elu. Nke a dị elu na pyramid na-akpọ apothem. Ya designation - "A". The n'ozuzu usoro nke ebe mpụta n'elu bụ dị ka ndị:

S = ½ P * A, ebe P - perimeta nke isi nke pyramid.

E nwere mgbe ụfọdụ ọ na-amaghị isi n'akụkụ, ma n'akụkụ n'ọnụ bụ (a) ewepụghị na n'akuku na onu ire (α). Mgbe ahụ, ọ na-adabere ná iji na-eso usoro gbakọọ mpụta ebe nke pyramid:

S = n / 2 ² mmehie α.

Task № 1

Ọnọdụ. Chọta ngụkọta nke ebe pyramid, ma ọ bụrụ na ya isi bụ ihe equilateral triangle na a n'akụkụ nke 4 cm na nwere uru √3 apothem cm.

Mkpebi. Ọ ga-amalite na na ngụkọta oge nke isi perimeta. Ebe ọ bụ na nke a bụ a mgbe nile triangle, mgbe ahụ, P = 3 4 = 12 cm apothem Dị ka a maara, onye pụrụ ozugbo gbakọọ ebe dum mpụta n'elu :. ½ 12 * √3 = 6√3 ^cm2.

Iji nweta ndị isi triangle bụ uru nke ebe (4 ² * √3) / 4 = 4√3 ^cm2.

Iji chọpụta dum ebe mkpa ka n'ogige atụrụ abụọ n'ihi ụkpụrụ: 6√3 + 4√3 = 10√3 ^cm2.

Azịza. 10√3 ^cm2.

Nsogbu № 2

Ọnọdụ. E nwere mgbe niile quadrangular pyramid. The ogologo nke isi bụ hà 7 mm, mpụta onu - 16 mm. Mkpa ka ị mara ya n'elu ebe.

Mkpebi. Ebe ọ bụ na polyhedron - akụkụ anọ na ezi na, na ya isi bụ a square. Ịnụ isi ebe na mpụta n'akụkụ-enwe ike ịgụ na square pyramid. The usoro maka square e nyere n'elu. Na m maara akuku nile ihu nke triangle. Ya mere, i nwere ike iji Heron si usoro maka ịgbakọ ha ebe.

The mbụ mgbawa ndị dị mfe na-eme ka a na nọmba: 49 mm ^2. Gbakọọ abụọ uru mkpa semiperimeter: (7 + 16 * 2): 2 = 19.5 mm. Ugbu a, anyị nwere ike gbakọọ ebe nke ihe isosceles triangle: √ (19,5 * (19,5-7) * (19,5-16) ²⁾ = √2985,9375 = 54.644 mm ^2. E nwere ihe anọ triangles, n'ihi ya, mgbe ịgbakọ ikpeazụ nọmba ga-mkpa mụbaa site 4.

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

Azịza. 267,576 chọrọ uru nke ² mm.

Task № 3

Ọnọdụ. Na mgbe quadrangular pyramid dị mkpa iji gbakọọ ebe. Ọ mara n'akụkụ nke square - 6 cm na ịdị elu - 4 cm.

Mkpebi. Ihe kacha ụzọ iji usoro iji ngwaahịa nke perimeta na apothem. The mbụ uru na-hụrụ naanị. Nke abụọ a nta ike.

Anyị ga-echeta Pythagorean Theorem ma tụlee a nri triangle. Ọ na-kpụrụ site elu nke pyramid na apothem, nke bụ hypotenuse. Nke abụọ ụkwụ bụ ọkara n'akụkụ nke square, dị ka a polyhedron elu dara n'etiti ya.

Mmasị apothem (na hypotenuse nke a nri triangle) bụ hà √ (March ² + 4 ²⁾ = 5 (cm).

Ugbu a, ọ bụ omume na-gbakọọ chọrọ uru: ½ * (4 6) * 5 + 6 ² = 96 (cm ^2).

Azịza. 96 cm ^2.

Nsogbu № 4

Ọnọdụ. Dana mgbe hexagonal pyramid. The n'akụkụ nke ya isi hà 22 mm, mpụta n'ọnụ - 61 mm. Gịnị bụ ebe mpụta elu nke a polyhedron?

Mkpebi. The echiche na ya bụ otu dị ka a kọwara na ọrụ №2. Naanị pyramid e nyere n'ebe ahụ square ke ukot, na ugbu a, ọ bụ a hegzagon.

Nzọụkwụ mbụ na gbakọọ site na isi nke ebe n'elu usoro ^{(6 22 2) / (} 4 * tg (180º / 6)) = 726 / (tg30º) = 726√3 ^cm2.

Ugbu a mkpa ka ị chọta ọkara perimeta nke ihe isosceles triangle, nke bụ a n'akụkụ ihu. (22 + 61 * 2) :. = 72 cm 2 anọgide na Heron si usoro gbakọọ ebe nke ọ bụla n'ime ndị triangle, wee ba uba ya site isii n'ogige atụrụ na onye na tụgharịa si na isi.

Mgbawa na Heron si usoro: √ (72 * (72-22) * ( 72-61) 2) = √435600 = 660 cm ^2. The mgbawa ga-enye mpụta n'elu ebe: 660 6 = 3960 cm ^2. Ọ na-anọgide tinye ha chọpụta dum elu: 5217,47≈5217 cm ^2.

Azịza. Grounds - 726√3 cm ^2, n'akụkụ elu - 3960 cm ^2, dum ebe - 5217 cm ^2.