Question

O piramida hexagonala regulata are latura bazei de 6 cm si volumul de 54 radical din 3 cm^3.
a) Aflați lungimea inaltimii piramidei
b) Determinati aria totala a piramidei

Answer 1

$V=\frac{A_b\times h}{3} \\\\A_b=\frac{3l^2\sqrt{3} }{2} \\\\A_l=\frac{P_b\times a_p}{2}$

$A_b=\frac{3\times36\sqrt{3} }{2} =54\sqrt{3}cm^2$

$54\sqrt{3} =\frac{54\sqrt{3} \times h}{3} \\\\h=3\ cm$

• Notam apotema bazei=$a_b$

• Notam apotema piramidei=$a_p$

$a_b=\frac{l\sqrt{3} }{2} =3\sqrt{3} cm$

• Aplicam Pitagora in triunghiul format din inaltimea piramidei, apotema bazei si apotema piramidei, ipotenuza fiind apotema piramidei:

$a_p^2=a_b^2+h^2$

$a_p^2=27+9=36\\\\a_p=6\ cm$

$P_b=6l=36cm\\A_l=\frac{36\times 6}{2} =108cm^2$

$A_t=A_l+A_b\\\\A_t=108+54\sqrt{3}$