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Dana takes a sheet of paper, cuts a 120-degree 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.

  • math -

    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

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