"An equilateral triangle of side length 6 sqrt 3 is rotated about an altitude to form a cone. What is the volume of the cone?"
Please don't solve the entire thing. A hint is all I really want so I can solve it by myself. Thanks!
radius of base = 3√3
height^2 + (3√3)^2 = (6√3)^2
find the height,
now you have everything needed in the formula for the volume of a cone
Ans is 81pi
Asking for a hint isn't cheating.
yes but aops would prefer it if they used their own website's message boards
To find the volume of the cone formed by rotating the given equilateral triangle, you can think about the process step by step.
First, visualize the equilateral triangle with side length 6√3. Draw one of its altitudes, which is also the height of the cone. Now, imagine rotating the triangle around this altitude to form a cone.
To find the volume of the cone, you need to know its base area and height. The base of the cone is the equilateral triangle, so you need to calculate its area.
To find the area of an equilateral triangle, you can use the formula:
Area = (side length^2 * √3) / 4
Once you have the base area, you need to determine the height of the cone. The altitude of the triangle is given as the height of the cone.
Finally, you can use the formula for the volume of a cone:
Volume = (base area * height) / 3
Keep in mind that this is a generalized hint to guide you through the problem-solving process. To calculate the numerical value using these formulas, perform the necessary calculations with the given data.