2pie r or pie r squared?

[soul58uk]

My son believes implicitly in the formula for circumferences whichever one it is!! And we have this argument.  

If you laid a rope around the equator it would have to be X'000 miles long.   If you then put the rope on 6 foot high poles around the equator it would need to be longer. The crux of the argument is that he insists that accourding to the mathematical formula it would only need to be about 18 feet longer I think, whereas I cannot accept this, logic to me dictates that it must have to be longer than that!!

Anyone got any better ideas than just relying on a mathematical formula?  

[moose!]

Bizarre but after scribbling on a piece of paper it seems its true.

 

Circumference = 2 X PI X radius

So with poles = 2 X PI X (radius + 6)

    = (2 X PI X radius) + (2 X PI X 6)

    = (2 X PI X radius) + (2 X 3-ish X 6)

    = (Original value) + 36

So I guess I make it 36 feet but still not much extra!

[moose!]

I guess you can compare this with putting a string around a beach ball.  To make a gap of a just a single millimeter around the beach ball wouldn't really need any noticeable increase in the amout of string.  So, in comparison, 6 feet isn't really much compared with the Earth's radius of 6000km.

[bernardo]

It would have to be 2 pi 6 feet longer, which is 12 pi which is about 38 feet.

 

This question is usually quoted as being a rope which is 6 feet longer, and which can be lifted almost a foot above the ground.

The circumference of a circle is 2 pi r, not pi r.

[bernardo]

by the way, the thing which "dictates" to you that it "must have to be" longer is not logic, but instinct.  This is where maths and logic are different from intuition and instinct.

[bernardo]

I have just realised that moose!'s answer is confused.  The amount of length that you need to increase the rope in order to produce an extra 6 feet of radius is 38 feet, regardless of how big the Earth is.  An extra 38 feet of length in a string round a beach ball (or a grain of sand, or the sun, or a ferris wheel) would produce an extra 6 feet of radius in all cases.

[newtron]

Yep, Bernardo's right.  That's the whole point of the problem.  The original radius "r" cancels out.

[moose!]

I realise that bernado, I was just trying to give a non-mathematical answer why such a small amount extra is required.

 

I don't think anyone would be surprised if you needed a lot of extra rope if you but 6-foot poles on a beach ball, now would they!

[magicdice]

Anyone got any better ideas than just relying on a mathematical formula?

*chokes on her fruit juice* just mathematics?!

sweetie if we didn't exist, there'd be no human logic, no instinct, no intuition. but zero would still exist, the formula for the circumference of a circle would be the same... mathematics is the one true thing in life that you genuinely can trust because it is unaffected by us: we don't theorise, experiment, invent, we discover mathematics. it's more true than anything you can possibly imagine

[Kingaroo]

"Pi r squared" is the AREA of a circle, not the circumference.
You can remember this because the answer is in SQUARE feet :(or whatever unit)

"2 pi r" (the correct formula for circumference) is also the same as "pi d" (D = diameter, which is twice the radius)
If you remember "pi D", you won't get confused by the number 2 being in both formulae.

[PatTheRat]

One little thing all you guys are forgetting here...2.pi.r is the formula for calculating the circumfrence of a circle, a perfect circle. Now correct me if I'm wrong, but i dont believe the earth is a perfect round shape. Hence 2.pi.r is too general to be used.

[magicdice]

PatTheRat you are correct, it's what is known as an oblate spheroid (i.e. it's a sphere that's slightly squished at the poles) however it's close enough to suffice for a theoretical discussion such as this, and the earth is just being used here as an illustrative device - it's not actually about the equator of the earth but about circumferences

[georgeous]

if you added another 6 ft to the radius of the Earth then the circumference would only be 1.915 metres longer to be precise

[georgeous]

sorry 1.9151

[landie]

I think you're mixed up gorgeous, the amount of 6ft and 1.9m (1.8288 metres actually) are the same thing. You wold need to multiply by 2 pi r to find the circumference, as others have rightly said.

[Hamish]

A non-mathamatical answer .... and experiment!

Draw a scale diagram of the earth and measure its circumfrence - ass the poles and draw in the rope - re measure the circumfrence. work out the differance (sorry maths here, but only arithmatic). Multiply by scaling facor - I think you will find the answer is quite small.

Note; 2pi r = pi d = circunfrence of circle

pi r^2  area of circle

[jimmydean]

An extra 38 feet of length in a string round a beach ball (or a grain of sand, or the sun, or a ferris wheel) would produce an extra 6 feet of radius in all cases.

Sorry but if you think about it.. this definatly isnt true.. If you had a beach ball with a 6 foot pole on it, you wouldnt need 38 feet of rope to go around it.. thats just redicilous.

You are treating the problem as if you are still working with a sphere, however you are not. when you add the 6 foot pole to the sphere you just changed the shape of the sphere to something else, and therefore the problem becomes more complex. there isnt just a simple solution to it.

[mibn2cweus]

Pi is the ratio of the circumference to the diameter, (not the radius which is only half the diameter). Extending the radius by a given amount all the way around would increase the diameter by twice that amount, in this case 12 feet. Remember the circumference is proportional to twice the radius . . . times pi.