math
posted by hodna on .
Dana takes a sheet of paper, cuts a 120degree circular sector from it, then rolls it up
and tapes the straight edges together to form a cone. Given that the sector radius is 12
cm, find the height and volume of this paper cone.

The circumference of the whole circle from which you cut the sector is 2π(12) = 24π cm
so using ratios to find the circumference of the sector:
120/360 = x/24π
x = 8π
That becomes the circumference of the base of the cone
for its radius r :
2πr = 8π
r = 4 cm
The radius of the original sector becomes the slant side of the cone
so h^2 + 4^2 = 12^2
h = √128 = 8√2
vol = πr^2h/3 = π(4)^2(8√2)/3 = (128π√2)/3 cm^3 or appr. 189.563