Thread: Volume of a Cone given the radius and lateral surface area

a 120° sector is cut out of a circular piece of tin with radius 6 in. and bent to form the lateral surface area of a cone. What is the volume of the cone?

I assume that the radius of 6 inches becomes the slant height but what am I supposed to do to find everything else needed to find the volume? What does the 120 degrees cut out have to do with it? Please explain this for me, the test is tomorrow and my brother the tutor isn't home tonight
Thanks!

Consider the circumference of the original materials and compare it to the circumference of the base of the cone. They are related and you will need the 120º for that.

Ah, I see. So if I understand what you're saying, I have to set up a proportion. 12\pi / x = 240/360
So the circumference of the cone is 56.53, right?

12*pi = 38ish How did it manage to get bigger by cutting out a piece?

Don't EVER forget to have something in mind before you start. You must know what is reasonable. Don't quite and think you are done when all you have is an unreasonable result.

Try x/(12pi) instead of (12pi)/x.

Plus, it's the circumference of the base of the cone.

Ok. When I do it that way, I get 25.13 for the circumference of thebase of the cone which makes more sense. What I don't understand is why the proportion is set up that way. I alwas thought that the terms on top referred to one thing and the terms on the bottom refer to the other thing. But the two terms on top, x and 240, are referring unknown circumference of the cone's base and the measure of the original circle. Why is that?

You do have to be consistent. 12pi is the whole thing. x is the reduced piece. Similarly, 360 is the whole thing and 240 is the reduced piece. You just got one of them backwards.

Oh, I get it now. I didn't think about it like that before. So now if the circumference of the cone's base is 25.13, then I divide by pi to find the diameter and then by 2 to find the radius which is about 4 (3.999). So the area of the base is 16pi. I uses the Pythagorean theorem to get the height because I know the slant height is 6 and the radius is 4. 36-16 is 20 so the height is the sqr. rt. of 20 times the base which is 16pi. That comes out to be 224.79. Divided by 3 is 74.93. Does that sound right?

Did you have a good estimate in mind before you started?

I would prefer the exact answers.

1) Circumference of base is 8pi
2) Diameter is 8
3) Radius is 4
4) Area of base is 16pi
5) Height is 2*sqrt(5)
6) Volume is (32/3)*pi*sqrt(5) = 74.931ish

No, I did not. But that's good advice. I usually make a lot of dumb mistakes and that would prevent a lot of them. Thanks for all the help!