# Length Of A Wire

1. Feb 6, 2007

### danago

1. The problem statement, all variables and given/known data
There are n identicle cylinders attached upright on a board. Each cylinder has a radius r, and the centre of each cylinder's base lie on a large circle of radius R. They are evenly spaced around the circle. A loop of wire encloses the cylinders. Show that the length of the wire is given by $$2\pi r + 2nRsin\frac{\pi}{n}$$

3. The attempt at a solution
Im quite lost. Im not even sure where to start. Im not asking for someone to show me the solution, but if somebody could start me in the right direction, that would be great.

Now, in class, we havnt started on limits yet, but i think they may be involved in this.

2. Feb 6, 2007

### AlephZero

Draw a picture, for a small values of n like 3 or 4.

The wire is wrapped tight round the cylinders so some parts of it are straight, and other parts are curved round the small cylinders.

You can get some clues from the answer. Try to see how the "2n angles of size pi/n" and the length "2 pi r" relate to your picture.

3. Feb 9, 2007

danago, your assumption of this involving limits is correct because as n approaches infinity the length of the wire circumnavigating the cylinders approaches 2(pi)(R+r)

The first part of the formula is straight forward, as the 2(pi)(r) indicates the wire length if it were around a single cylinder.
When having drawn your diagrams, you will have noticed that triangles can be drawn between the area enclosed by the wire, thus the furthermost edge of the triangles plus the 2(pi)(r) will give you the length of the wire. To calculate the furthermost edge of the triangles try using the sine rule (a/sin A = b/sin B = c/sin C)