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Giải mục 1 trang 80 SGK Toán 11 tập 2 - Cùng khám phá

Viết công thức tính thể tích khối lăng trụ đứng tam giác (Hình 8.68)
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Hoạt động 1

Viết công thức tính thể tích khối lăng trụ đứng tam giác (Hình 8.68) và khối lăng trụ đứng tứ giác (Hình 8.69) theo diện tích đáy S và đường cao h của nó.

Phương pháp giải:

V = S.h

Lời giải chi tiết:

Công thức tính thể tích khối lăng trụ đứng: V = S.h

Luyện tập 1

Cho hình lăng trụ ABC.A’B’C’ có đáy ABC là tam giác đều cạnh a , góc giữa cạnh bên và mặt đáy bằng 300🌜. Hình chiếu của A’ trên (ABC) là trung điểm I của cạnh BC. Tính thể tích khối lăng trụ này.

Phương pháp giải:

Công thức tính thể tích khối lăng trụ: V = S.h S: diện tích đáy, h: chiều cao

Lời giải chi tiết:

Gọi I là trung điểm BC \(AI = \sqrt {A{C^2} - I{C^2}}  = \sqrt {{a^2} - {{\left( {\frac{1}{2}a} \right)}^2}}  = \frac{{\sqrt 3 }}{2}a\) Tam giác AB đều nên AI vuông góc với BC hay AI vuông góc với (ABC) Suy ra \(\widehat {\left( {AA',\left( {ABC} \right)} \right)} = \widehat {A'AI} = {30^0}\) I là hình chiếu của A’ trên (ABC) nên A’I vuông góc với BC Suy ra tam giác A’AI vuông tại I có:  \(\tan {30^0} = \frac{{A'I}}{{AI}} = \frac{{A'I}}{{\frac{{\sqrt 3 }}{2}a}} \Rightarrow A'I = \frac{1}{2}a\) \(V = {S_{\Delta ABC}}.A'I = \frac{1}{2}AI.BC.A'I = \frac{1}{2}.\frac{{\sqrt 3 }}{2}a.a.\frac{1}{2}a = \frac{{\sqrt 3 }}{8}{a^3}\)

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