The volume of the cone is (1/3)πr²h. We are given that the radius is 12 and the volume is 432π, so we can solve for the height:
(1/3)π(12²)h = 432π h = 4
The slant height of the cone is equal to the radius of the base, so it is 12 cm. The height of the cone is the difference between the slant height and the radius, so it is 12 - 4 = 8 cm.
The unshaded sector is a sector of a circle with radius 12 and central angle 360 - 8 = 272 degrees. The shaded sector is a sector of a circle with radius 12 and central angle 360 - 4 = 356 degrees.
The ratio of the shaded sector's central angle to the unshaded sector's central angle is equal to the ratio of the shaded sector's area to the unshaded sector's area. The area of a sector is (central angle/360)πr², so the ratio of the shaded sector's area to the unshaded sector's area is
(356/272)π(12²) = (356/272)(144)π = 21(12)π
The ratio of the shaded sector's central angle to the unshaded sector's central angle is 21/16, so the unshaded sector's central angle is 16/21 * 360 = 256 degrees. Therefore, the measure of angle ABC of the sector that is not used is 360−256=104 degrees.
