## How Long Is A Piece Of String?

Posted by Karl Sharman on May 24, 2011 – 11:12 am

A nice easy one to relieve the midweek brain-strain…

A cylinder 84 cm high has a circumference of 16 cm. A string makes exactly 7 complete turns round the cylinder while its two ends touch the cylinder’s top and bottom. How long is the string?

I want to see the working out behind this too!

May 24th, 2011 at 12:08 pm

140 cm.

I’ll break it down into its parts:

First, 1 time around ends up 12 cm higher than the start point (84/7=12). Second, it goes 16 cm around.

These are the 2 sides of a right triangle if you were to “unroll” the cylinder. (12=height, 16=base)

So sqrt(12^2 + 16^2) will give us a length of 20, which is the straight-line path the string will take.

Now simply multiply that by 7 wraps, and you get 140 cm.

May 24th, 2011 at 12:14 pm

140 cm I agree with DP assuming the string takes the shortest path.

I used a similar method as DP except I used the cylinder’s original 84 cm height and multiplied the 16cm circumference by 7 to get 112cm circumference. I unrolled the cylinder and had the string complete the hypotenuse of the triangle.

84^2 + 112^2 = 140^2

May 24th, 2011 at 12:15 pm

140cm

the same solution, I just “open” the cylinder to draw the string’s path.

May 24th, 2011 at 12:46 pm

Hi Karl. I nearly posted this one myself.

May 24th, 2011 at 1:54 pm

You didn’t say how big the string was or the overlap of the touch on each end.

Assume the string is 12 cm in diameter, then as you wrap the string around the cylinder, the length is 7*16 or 112, given the slop of where such a large string “touches”. If there is some argument here about touching, after you have measured this string out, cut it into 7 equal lengths, wrap each length with first touching last and side of next touching side of last and again it will still be 112.

But I actually like what DP said.

May 24th, 2011 at 3:59 pm

N = L / 2 ⁄7 r

7 = L / 2*(22/7)*8

L = 352 cm

May 24th, 2011 at 7:38 pm

I also came up with John24’s method.

May 24th, 2011 at 7:44 pm

Me too. 140 cm.

May 25th, 2011 at 9:30 am

TheWyvern – unlike you to get this wrong? DP gets in very quickly with the right answer, and as most have said – “unroll” the cylider for an easy view of the problem!

NewMan – The ends of the string were touching – no mention of overlap.

May 31st, 2011 at 5:04 am

it dos not

May 31st, 2011 at 8:34 am

Definitely 140 cm.

October 27th, 2012 at 11:19 am

80 cm

August 19th, 2014 at 2:01 am

140 is the wrong answer…ppl….

August 19th, 2014 at 2:39 am

if the question is other way round that there is a cylinder of 15 cm and 4cm is the circumference of the cylinder and a string is required to wrap around it at the interval of 3 cm so that whole cylinder is complete then in that case what will be your approach to solve the problem ?

August 19th, 2014 at 8:54 am

Hi Apollo. Why do you say 140 cm is the wrong answer? By unwrapping the cylinder we can see that there’s a right angled triangle of base 7*16 = 112 cm and height 84 cm. So the hypotenuse is 140 cm.

If I’ve understood your question correctly, the string wraps around the cylinder 15/3 = 5 times. So the length of the string is sqrt((5*4)^2 + 15^2) = 25 cm