How do I calculate the length of a circle when the circle is stretched out to a line?

[gyrex]

Hi,

I have an interesting question which has arisen from an everyday problem. I have a square bin which has 42cm sides and measures ~52cm diagonally.

I am trying to calculate and buy garbage bags which would fit the bin. Obviously garbage bags are circular when stretched out but the measurements given on any given package are the dimensions of the bag when it is stretched out from point to point. It would be easy if the measurements on the packets were expressed in diameter because then I could buy 52cm diameter garbage bags, but they're not so here I am...

I guess what I'm trying to calculate is if you have a circle that you can manipulate and stretch so that it now forms a line, what would be the length of that line?





Now, I could just buy a bunch of garbage bags and use trial and error but that's too easy and the geek in me would like to actually work out a formula on how to calculate the required garbage bag dimensions.

IMO interesting question nonetheless and it'd be fantastic if there's some formula to calculate this?

Cheers,

John

 

[gyrex]

I think I may have worked out the answer but if someone could double check my theory, I'd really appreciate it...

Work out the circumference of the desired circle that would fit the square bin, using the dimensions above: 52 x pi = ~164cm. Divide that by 2 and we get the circle stretched into 2 opposing lines which = 82cm?

Please let me know if my theory is flawed.

 

[berni2723]

hi, first of all, the bag will only be circular if you wish to shape it that way. the bottom of the bag may be finished in a way so that it tapers to a round-like bag, but the top can be shaped in any way, even zig-zag...if that is the shape of the container.

but to answer the math problem...

as you can see from your (great) diagram, the length you are looking at when flattening the circle is simply half the circumference. therefore, if it is for example 2 feet long, then the circumference of the bag when open will be 4 feet around. to find the diameter, well circumference = 2 x pi x radius of diameter x pi , so dividing your circumference by pi will give you the diameter. hope that helps.

 

[berni2723]

I think I may have worked out the answer but if someone could double check my theory, I'd really appreciate it...

Work out the circumference of the desired circle that would fit the square bin, using the dimensions above: 52 x pi = ~164cm. Divide that by 2 and we get the circle stretched into 2 opposing lines which = 82cm?

Please let me know if my theory is flawed.

good work, but as i mentioned in the reply i wrote before reading this, if all sides of the bin are 42cm, then the perimeter is 168cm.

now, just as the bag can be stretched and shaped to a closed form of 2 sheets of plastic, you can simply get a bag which is 84cm when closed, and fit it on the bin

 

[Quaid]

Work out the circumference of the desired circle that would fit the square bin, using the dimensions above:

52 x pi = ~164cm. Divide that by 2 and we get the circle stretched into 2 opposing lines which = 82cm?

Please let me know if my theory is flawed.

You stated that 52 cm is the diagonal of the square, and that each side of the square is 42 cm. Those numbers don't jive.

A 42-centimeter square has a diagonal of 59.4 cm (rounded).

Also, if the diameter of the circle is 42 cm, then its radius is 21 cm.

The circumference of a circle is 2×Pi×radius, so yours is (2)(3.1416)(21).

Half of that works out to 66 cm (rounded).

How did you get 52 cm, for the diagonal of the square?

Cheers :cool:

PS: Not certain about the "stretching" part ...

 

[gyrex]

You stated that 52 cm is the diagonal of the square, and that each side of the square is 42 cm. Those numbers don't jive.

A 42-centimeter square has a diagonal of 59.4 cm (rounded).

Also, if the diameter of the circle is 42 cm, then its radius is 21 cm.

The circumference of a circle is 2×Pi×radius, so yours is (2)(3.1416)(21).

Half of that works out to 132 cm (rounded).

How did you get 52 cm, for the diagonal of the square?

Cheers :cool:

PS: Not certain about the "stretching" part ...

Thanks for your replies!

Firstly, I should have pointed out that the bin had rounded corners but I wasn't too fussed about this, I was just trying to find a formula that fits my purpose.

Secondly, the bin measures 52cm on a diagonal. I can't use 42cm (being the length of each side of the square) per your response because the circle wouldn't have a large enough diameter to fit around the bin at the diagonal, thus I need to use the diagonal as a diameter for the circle.

I'm happy with my calculations and that a garbage bag with top measurement of 82cm will be appropriate for my needs

Thanks again guys!

 

[gyrex]

The circumference of a circle is 2×Pi×radius, so yours is (2)(3.1416)(21).

Half of that works out to 132 cm (rounded).

FYI, I could be wrong but I think your maths is out here. I think your calculation should be (42)(pi) = 132 / 2 = 66cm?

 

[gyrex]

I'm pretty sure you guys are using the wrong dimension for the circle diameter here. Per the diagram below, I'm proposing to use the diameter of the diagonal (in red) and you guys are using the length of the side of the square (in blue) which as you can see, wouldn't form a large enough circle to fit the (rounded) square.

 

[Quaid]

FYI, I could be wrong but I think your maths is out here. I think your calculation should be (42)(pi) = 132 / 2 = 66cm?

You are correct! I copied the wrong number.

Also, I was considering the inner circle because I did not realize that you were stretching the bag around the outside of the bin.

Glad you figured it out...