Triangular prism,

The regular triangular prism, whose edges are identical, has a surface of 2514 cm ^ 2 (square). Find the volume of this body in cm3 (l).

Result

V =  7180.447 cm3

Solution:

S=2514 cm2 S1=3/4 a2  S=2⋅ S1+3a⋅ a  S=3/2 a2+3a2 S=(3/2+3)a2  a=S3/2+3=25143/2+3≐25.5006 cm  S1=3/4⋅ a2=3/4⋅ 25.50062≐281.5796   V=S1⋅ a=281.5796⋅ 25.5006≐7180.4472≐7180.447 cm3   Correctness test:  S2=2⋅ S1+3⋅ a2=2⋅ 281.5796+3⋅ 25.50062=2514 cm2 S2=SS=2514 \ \text{cm}^2 \ \\ S_{1}=\sqrt{ 3 }/4 \ a^2 \ \\ \ \\ S=2 \cdot \ S_{1} + 3a \cdot \ a \ \\ \ \\ S=\sqrt{ 3 }/2 \ a^2 + 3a^2 \ \\ S=(\sqrt{ 3 }/2 +3 )a^2 \ \\ \ \\ a=\sqrt{ \dfrac{ S }{ \sqrt{ 3 }/2 +3 } }=\sqrt{ \dfrac{ 2514 }{ \sqrt{ 3 }/2 +3 } } \doteq 25.5006 \ \text{cm} \ \\ \ \\ S_{1}=\sqrt{ 3 }/4 \cdot \ a^2=\sqrt{ 3 }/4 \cdot \ 25.5006^2 \doteq 281.5796 \ \\ \ \\ \ \\ V=S_{1} \cdot \ a=281.5796 \cdot \ 25.5006 \doteq 7180.4472 \doteq 7180.447 \ \text{cm}^3 \ \\ \ \\ \text{ Correctness test: } \ \\ S_{2}=2 \cdot \ S_{1} + 3 \cdot \ a^2=2 \cdot \ 281.5796 + 3 \cdot \ 25.5006^2=2514 \ \text{cm}^2 \ \\ S_{2}=S

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