This is a question that needs to be visualized.
(I used to go to our teacher's staff room and bring back a cone paper cup from the water cooler and cut it open)
One has to realize that the arclength of the sector becomes the circumference of the circle of the cone, and the original radius of the sector becomes the slant height of the cone.
so for a 180° sector arclength = (1/2)(2π)12 = 12π cm
so the circumference of the base circle of the cone = 12π
2πr = 12π and the radius of the cone base is 6
then the height h is ...
h^2 + 6^2 = 12^2
h = √108 = appr. 10.3923
Volume of cone = (1/3)πr^2h = (1/3)π(36)√108 = 391.78
since 120° is 1/3 of the rotation, the arclength would be (1/3) of 2π(12) or 8π
find the radius of the base circle as above, then the height, and finally the volume.
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