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ADVANCED AMC8 GEOMETRY QUESTION/MATH

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An equilateral triangle of side 14 centimeters is revolved about an altitude to form a cone. What is the number of cubic centimeters in the volume of the cone? Express your answer to the nearest whole number, without units.

I just want the answer. Thanks! :)

  • ADVANCED AMC8 GEOMETRY QUESTION/MATH -

    Note that an equilateral triangle is a triangle where all lengths of the sides are equal.
    If you revolve the triangle about its altitude (or the height), you'll generate a cone (just try to imagine or draw the figure). Therefore, the height of the cone is equal to the height of triangle, and its radius is equal to half of one side of the equilateral triangle, which is 14/2 = 7 cm (radius).
    To get the height, use pythagorean theorem:
    c^2 = a^2 + b^2
    where
    c = hypotenuse (in this case, 14 cm)
    a = height of triangle (which is also cone height)
    b = base (which is also the cone radius)
    Substituting,
    14^2 = a^2 + 7^2
    a^2 = 14^2 - 7^2
    a^2 = 147
    a = 7*sqrt(3)
    Finally, we get the volume of cone. Recall that the volume of cone is just
    V = (1/3)*(pi)*(r^2)*h
    Substituting,
    V = (1/3)(3.14)(7^2)(7*sqrt(3))

    Now solve for V. Units in cubic centimeters.

    Hope this helps~ :)

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