cone

The   slant height  of  the cone   will  =  the  height of  the  equilateral triangle

This  slant height  =  sqrt  [ 6^2 - 3^2  ]  =  sqrt  [ 36 - 9 ]  =  sqrt (27 ) 

The  circumference of the  cone  = side of the  triangle = 6

We can find the radius, r, of the  cone as

circumference =  2pi *r

6 = 2pi r

r =  6/ (2pi)  =   3/pi

Heigt of  cone =   sqrt  [  [sqrt (27)]^2  - (3/pi)^2  ]   =  sqrt [ 27 - 9/ pi^2 ]  =  sqrt  [ 27pi^2 - 9]  / pi  =

sqrt  [ 9 (3 pi^2 - 1 ) ] /pi  =     (3/pi) sqrt ( 3pi^2 - 1 )

Volume of  cone =  (1/3) radius^2  * height   =   

(1/3)  ( 3/pi)^2 *  (3/pi) sqrt ( 3pi^2  - 1) =

(9/ pi^3)sqrt (3pi^2 -1) ≈   1.55 units^3

An equilateral triangle of side length 6 is rotated about an altitude to form a cone. What is the volume of the cone?

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Cone radius         r = 3 

Slant height         h_s = 6               ( Height  h = √27 )

Cone volume         V = 1/3 * pi * r² * sqrt(h_s² - r²)    ==>  V = 48.97258284 u³

                                         or

                              V = r² * pi * h / 3        ===> V = 48.97258284 u³