Пирамида

[Гост]

Правилна четириъгълна пирамида има височина h и ъгъл 2[tex]\alpha[/tex] между две съседни околни стени.Намерете околната и повърхнина.Моля за помощ.

[ptj]

пирамида [tex]ABCDV[/tex]; [tex]AB=a[/tex]; т.[tex]O\in BV: (AOC)\bot VB[/tex]

[tex]\frac{AC}{2}=\frac{\sqrt{2}.a}{2}=AO.sin\alpha\Rightarrow AO=\frac{\sqrt{2}.a.sin\alpha}{2}[/tex]

[tex]sin(\angle ABO)=\frac{AO}{AB}=\frac{\sqrt{2}.sin\alpha}{2}[/tex]

[tex]VV_1[/tex] - апотема в [tex]\Delta ABV[/tex]

[tex]VV_1=\frac{AB}{2}.tg(\angle ABO)=\frac{a}{2}.\frac{\frac{\sqrt{2}.sin\alpha}{2}}{\sqrt{1-(\frac{\sqrt{2}.sin\alpha}{2}})^2}=\frac{a.sin\alpha}{2\sqrt{2-sin^2\alpha}}[/tex]

[tex]V_{ABCV}=\frac{OC.S_{ABV}}{3}=\frac{OC.AB.VV_1}{6}=\frac{\frac{\sqrt{2}.a.sin\alpha}{2}.a.\frac{a.sin\alpha}{2\sqrt{2-sin^2\alpha}}}{6}=\frac{\sqrt{2}.a^3.sin^2\alpha}{24\sqrt{2-sin^2\alpha}}[/tex]

[tex]V_{ABCV}=\frac{h.S_{ABC}}{3}=\frac{a^2h}{6}[/tex]

[tex]a=\frac{2h\sqrt{4-sin^2\alpha}}{sin^2\alpha}[/tex]

[tex]S_{ok}=4S_{ABV}=2.AB.VV_1=\frac{a^2.sin\alpha}{\sqrt{2-sin^2\alpha}}=\frac{8h^2\sqrt{2-sin^2\alpha}}{sin^3\alpha}[/tex]