### Instruction: An equilateral triangle of side 12 centimeters is rotated about an altitude to form a cone. What is the number of cubic centimeters in the volume of the cone?### Response: To find the volume of the cone, I need to know its height and radius.
The height is the same as the altitude of the triangle, which I can find using the Pythagorean theorem on a 30-60-90 right triangle.
One leg of the right triangle is half of the side of the equilateral triangle, so 6 centimeters, and the hypotenuse is the full side, 12 centimeters.
The other leg is the altitude, and I can solve for it by squaring both sides of the equation 6^2 + x^2 = 12^2 and simplifying.
I get x^2 = 108, so x = sqrt(108), which I can simplify further by factoring out a perfect square: x = sqrt(36*3) = 6*sqrt(3).
So the height of the cone is 6*sqrt(3) centimeters.
The radius of the cone is also 6 centimeters, since it is the same as the leg of the right triangle.
Now I can use the formula for the volume of a cone, which is (1/3)*pi*r^2*h, where r is the radius and h is the height.
Plugging in the values I found, I get (1/3)*pi*6^2*6*sqrt(3) = 72*pi*sqrt(3) cubic centimeters.
This is the exact volume of the cone.