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I'm stuck on this

 

A right pyramid has a square base.  The area of each triangular face is one-third the area of the square face.   If the total surface area of the pyramid is $432$ square units, then what is the volume of the pyramid in cubic units?

 Oct 12, 2023
 #1
avatar+128393 
+1

Surface area  =   b^2  + 4 * ( b^2 /3)

432 = (7/3)b^2

b^2 = 432 ( 3/7)  = 1296 / 7

b = sqrt (1296/ 7)  = 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 * sqrt 7 / 18 =  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 = 

 

(1/3)  base area * height  =

 

(1/3) (b^2) * h

 

(1/3) (1296  / 7) * 6  =  2592 / 7 units^3   ≈ 370.29  units^3  

 

 

cool cool cool

 Oct 12, 2023

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