Kub va parallelepiped: matematika darsi

Xona, quti, g'isht: atrofdagi eng ko'p uchraydigan jismlar to'g'ri burchakli parallelepipeddir. O'lchamlari $3$ , $4$ va $12$ bo'lgan qutiga sig'adigan eng uzun tayoq $\sqrt{3^2+4^2+12^2}=13$ ga teng. Bu darsda parallelepipedning barcha turlari, ularning formulalari va kesim yuzini topish usullari beriladi.

Turlari

Jism	Ta'rifi	Hajmi
JismParallelepiped	Ta'rifibarcha $6$ yog'i parallelogramm	Hajmi $V=S_{\text{asos}}\cdot H$ , $H$ balandlik
JismTo'g'ri parallelepiped	Ta'rifiyon qirralari asosga perpendikulyar, asosi parallelogramm	Hajmi $V=S_{\text{asos}}\cdot H$ , $H$ yon qirra
JismTo'g'ri burchakli parallelepiped	Ta'rifibarcha yoqlari to'g'ri to'rtburchak	Hajmi $V=abc$
JismKub	Ta'rifibarcha yoqlari kvadrat	Hajmi $V=a^3$

Har qanday parallelepipedda $6$ ta yoq, $12$ ta qirra, $8$ ta uch bor. Qarama-qarshi yoqlari teng va parallel. To'rtta diagonali bir nuqtada kesishadi va shu nuqtada teng ikkiga bo'linadi. Og'ma parallelepipedda $H$ yon qirra emas: yon qirra $l$ asos bilan $\varphi$ burchak hosil qilsa, $H=l\sin\varphi$ . Masalan, asos yuzi $12$ , $l=6$ , $\varphi=30^\circ$ bo'lsa, $H=3$ va $V=36$ .

To'g'ri burchakli parallelepiped

V=abc

S=2(ab+bc+ac)

S_{\text{yon}}=2(a+b)c

a

b

asos tomonlari,

c

balandlik

d^2=a^2+b^2+c^2

Diagonal: barcha to'rtta diagonal teng

Pifagor teoremasi ikki marta qo'llanadi:

{AC^2=a^2+b^2}

, keyin

{AC_1^2=AC^2+CC_1^2}

Diagonal formulasining kelib chiqishi: $CC_1$ qirra asosga perpendikulyar, shuning uchun $ACC_1$ uchburchakda $C$ burchak to'g'ri. Yoq diagonallari: $\sqrt{a^2+b^2}$ , $\sqrt{a^2+c^2}$ , $\sqrt{b^2+c^2}$ .

S_1S_2S_3=V^2

Uch qo'shni yoq yuzlari:

{ab\cdot bc\cdot ac=(abc)^2}

Yoqlari yuzi $6$ , $10$ , $15$ bo'lsa, ${V=\sqrt{900}=30}$ , qirralar esa:

\frac{30}{15}=2,\ \ \frac{30}{10}=3,\ \ \frac{30}{6}=5

Diagonal va asos orasidagi burchak:

\operatorname{tg}\varphi=\frac{c}{\sqrt{a^2+b^2}}

Kub

Kattalik	Formula	$a=4$ uchun
KattalikHajm	Formula $V=a^3$	$a=4$ uchun $64$
KattalikTo'la sirt	Formula $S=6a^2$	$a=4$ uchun $96$
KattalikYon sirt	Formula $S_{\text{yon}}=4a^2$	$a=4$ uchun $64$
KattalikYoq diagonali	Formula $a\sqrt2$	$a=4$ uchun $4\sqrt2$
KattalikKub diagonali	Formula $a\sqrt3$	$a=4$ uchun $4\sqrt3$
KattalikDiagonal kesim yuzi	Formula $a\cdot a\sqrt2=a^2\sqrt2$	$a=4$ uchun $16\sqrt2$
KattalikUchdan u tegishli bo'lmagan diagonalgacha masofa	Formula $\dfrac{a\sqrt6}{3}$	$a=4$ uchun $\dfrac{4\sqrt6}{3}$

Oxirgi qator: $ACC_1$ to'g'ri burchakli uchburchakda $C$ dan gipotenuzaga balandlik:

\frac{a\sqrt2\cdot a}{a\sqrt3}=\frac{a\sqrt6}{3}

Kub diagonali bilan yoq orasidagi $\varphi$ burchak va diagonal bilan qirra orasidagi $\psi$ burchak:

\operatorname{tg}\varphi=\frac{a}{a\sqrt2}=\frac{1}{\sqrt2}

\cos\psi=\frac{1}{\sqrt3}

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