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