# Thread: Circular functions and trigonometry

1. ## Circular functions and trigonometry

The question is as follows:

The first diagram shows a circular sector of radius 10 cm and an angle of theta radians (not the angle of the cut out sector). The second diagram shows it formed into a cone of slant height 10 cm. the vertical height, h, of the cone is equal to the radius, r of its base. Find the angle theta in radians.

My reasoning is as follows:

10^2 = 2x^2
x= 7.07

V=1/3Pi 7.07^2*7.07
V=370 cm cubed

My reasoning was as follows the volume obtained would seem to be equal to the area of a volume of a cylinder - the volume of the sector cut out. Have I done something completely wrong?
V= Pi*r^2*h - 1/2 theta r^2 *h

2. Hi

To be clear : is it the situation ?

3. Almost except we are to find the size of the other angle in diagram one.

First of all, is my reasoning sound? If so does one take the found angle in radians and subtract it from 6.283 radians (approximately 360 degrees) to find the angle?

4. OK
I forgot to add that the radius of the circle in the first diagram is s

You should not reason on volumes but on areas
You can find $\theta$ by saying that the area of the circle (without cut sector) is equal to the area of the lateral side of the cone (formula is $\pi\:r\:s$)

5. Yeah I thought I was complicating the question a tad bit.

6. Can you conceive of any other method, because that formula is not part of the formula booklet that we've been granted. Is there any way to derive the formula from any other formulae?

7. I do not see any other way ...

8. Since area of the curved surface of a cylinder is A=2*Pi*r*h Would it make sense seeing the relationship between the volume of the two to take that and multiply it by 1/3? If not(which I am guessing is the case considering that it does not correspond with the formula you submitted), do you know how to derive the formula?