Question 1204261


Surface area : 

{{{ A=   b^2  + 4 * ( b^2 /3) }}}

{{{ A=   (7/3)b^2}}}


given that {{{ A=432}}}


{{{ 432 = (7/3)b^2}}}

{{{ b^2 =432/(7/3) =432 ( 3/7)  = 1296 / 7}}}

{{{ b = sqrt (1296/ 7)  }}}

{{{ b= 36 /sqrt (7)}}}


To find the  slant  height, {{{ s}}}, of the pyramid

area of face = {{{ (1/2)b*s}}}

{{{ b^2 / 3 =  (1/2) (36 / sqrt( 7))*s}}}

{{{ 1296 / (7*3)  = (18 / sqrt (7))*s}}}

{{{ s = (1296 / 21) /(18 / sqrt (7))}}}

{{{ s=  24 / sqrt (7)}}}


To find the height, {{{ h}}}, of the pyramid by the Pythagorean Theorem

{{{ h  =  sqrt  ( s^2  - (b/2)^2 )}}}

{{{ h =  sqrt ( (24/sqrt (7))^2  - (18/sqrt( 7))^2 )}}}

{{{ h =   6}}}
 

Volume of pyramid

{{{ V=(1/3) (b^2) * h}}}

{{{ V=(1/3) (1296  / 7) * 6 }}}

{{{ V=   (2592 / 7) }}}

{{{ V }}}≈ {{{ 370.29 }}}