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Bài 7.28 trang 63 SGK Toán 11 tập 2 - Kết nối tri thức

Cho khối chóp đều S.ABC, đáy có cạnh bằng a, cạnh bên bằng b. Tính thể tích của khối chóp đó.

🥃Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Cho khối chóp đều S.ABC, đáy có cạnh bằng a, cạnh bên bằng b. Tính thể tích của khối chóp đó. Từ đó suy ra thể tích của khối tứ diện đều có cạnh bằng a.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Thế tích khối chóp \(V = \frac{1}{3}h.S\)

Lời giải chi tiết

 

Vì hình chóp S.ABC đều, gọi G là hình chiếu của S trên (ABC) nên G là tâm của đáy ABC là tam giác đều do đó G cũng là trọng tâm hay trực tâm của tam giác ABC. Gọi AG cắt BC tại D Tam giác ABC đều cạnh a nên \(AD = \frac{{a\sqrt 3 }}{2}\) Mà G là trọng tâm nên \(AG = \frac{2}{3}AD = \frac{{a\sqrt 3 }}{3}\) Xét tam giác SAG vuông tại G có \(SG = \sqrt {S{A^2} - A{G^2}}  = \sqrt {{b^2} - {{\left( {\frac{{a\sqrt 3 }}{3}} \right)}^2}}  = \sqrt {{b^2} - \frac{{{a^2}}}{3}} \) Diện tích tam giác đều ABC là ${{S}_{\vartriangle ABC}}=\frac{{{a}^{2}}\sqrt{3}}{4}$ Thể tích khối chóp đều là $V=\frac{1}{3}SG.{{S}_{\vartriangle ABC}}=\frac{1}{3}.\sqrt{{{b}^{2}}-\frac{{{a}^{2}}}{3}}.\frac{{{a}^{2}}\sqrt{3}}{4}=\frac{{{a}^{2}}\sqrt{3}}{12}.\sqrt{{{b}^{2}}-\frac{{{a}^{2}}}{3}}$ Do đó thể tích của khối tứ diện đều có cạnh bằng a là

\(V = \frac{{{a^2}\sqrt 3 }}{{12}}.\sqrt {{a^2} - \frac{{{a^2}}}{3}}  = \frac{{{a^3}\sqrt 2 }}{{12}}\)

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