An equilateral triangle is rotated about one its altitudes, sweeping out a cone in space. If the side length of the equilateral triangle is 12√3 then find the volume of the cone.

Question

a thorough explanation would be great! I am a beginner in the higher geometry :)

Answer 1

V=(1/3)∏r^2h is the formula for the volume of a cone

the radius of the cone is 1/2 the side length of the equilateral triangle, or r=6√3

the height of an equilateral triangle is h=a√3/2 where a=side length

h=(12√3)(√3)/2

h=36/2=18 (you can find this using the Pythagorean Theorem if you didn't know this formula)

r=6√3 and h=18

substitute into the formula for the volume of a cone

V=(1/3)(∏)(6√3)^2(18)

V=(1/3)(∏)(108)(18)

V=∏*36*18

V=648∏