Calculating Relative Error for Circle Measurements: A Comprehensive Guide

[Ashley11]

Relative Error!?

If the relative error for the radius of a circle is 5%, what is the relative error for the circumference?

I know that relative error is the ratio of the absolute error in a measurement to the size of the measurement, but I honestly have no idea how to complete the problem. These are one of the pre-lab problems, where we're supposed to try and figure it out before we're taught it, but I'm stumped on this one.

PLEASE HELP!

 

[LowlyPion]

If the relative error for the radius of a circle is 5%, what is the relative error for the circumference?

I know that relative error is the ratio of the absolute error in a measurement to the size of the measurement, but I honestly have no idea how to complete the problem. These are one of the pre-lab problems, where we're supposed to try and figure it out before we're taught it, but I'm stumped on this one.

PLEASE HELP!

What is the formula for the circumference of a circle?

If r is off by 5%, what effect will that have on the relative error of the formula for circumference?

 

[dlgoff]

Use the equation for finding the circumference from the radius. Since the radius is the measured variable, only error from it will apply.

 

[Ashley11]

Well, I came up with two different answers using the circumference formula: 10% and 31.4%. I'm relatively uncertain if I've done this correctly, but I believe 31.4% to possibly be the correct answer. The problem that I'm stuck on seems to be the simplest!

 

[fluidistic]

The circumference of a circle is $$2\pi r$$. As dlgoff said, we assume there is only an error in $$r$$, so what's the error of $$2\pi r$$?
EDIT: I think you got it right, the error would be $$2\pi \cdot 5$$% which seems what you did.

 

[Ashley11]

Wait...since circumference is proportional to radius, would that make the relative error of the circumference 5%, as well?

 

[LowlyPion]

Well, I came up with two different answers using the circumference formula: 10% and 31.4%. I'm relatively uncertain if I've done this correctly, but I believe 31.4% to possibly be the correct answer. The problem that I'm stuck on seems to be the simplest!

Try it this way. Think about what the relative error means.

Doesn't it mean that the measured value can be as great as 5%?

That (r+.05r)/r yields an error of 5%

Now substitute that into the equation for the circumference.

C = 2πr

Now on the one hand the absolute error will be 2π greater, but then so would the nominal absolute result.

In calculating your RELATIVE error you compare then the Error Result with the magnitude of the correct result.

In that case it becomes (2π*(r+.05))/2π*r After simplifying though what are we left with? (r+.05)/r? And was that our original error in r?

So what does that say about multiplying Relative errors by a constant?

 

[LowlyPion]

Wait...since circumference is proportional to radius, would that make the relative error of the circumference 5%, as well?

There you go you beat me to it.

Good thinking.

 

[Ashley11]

Great! Thanks a lot for everyone's help! Great, GREATLY appreciated :)