Uncertainty in area of a circle

• zero13428
In summary, the radius of a circle is 14.3+-0.3cm and the formula for its area is A=(∏)(r)^2. To find the uncertainty in the area, we can take the partial derivative ∂A/∂r=2πr or ∂A=2πr ∂r and divide it by A, giving us ∂A/A = 2∂r/r. This can also be expressed as dA/A = 2*dr/r. By substituting the values given, the uncertainty in the area is calculated to be 26.9cm.
zero13428

Homework Statement

The radius of a circle is measured to be 14.3+-0.3cm. Find the circle's area and the uncertainty in the area.

I don't understand how to correctly apply uncertainty equations with sigma and partial derivatives to these types of problems.

Homework Equations

A=(pi)(r^2)
(pi)(r^2)=642.4cm

Well then, we have A=πr2. If we take ln of both sides we will get

lnA=ln(πr2)=lnπ+2lnr

Now just take differentials

dA/A = 2*dr/r

dA is nothing but the error in A. Same with dr. Just substitute the numbers.

I really could not explain it properly without showing you the differentials.

You said to take the ln of both sides. As in the natural log? I didn't know these had anything to logs or am I reading something wrong.

zero13428 said:
You said to take the ln of both sides. As in the natural log? I didn't know these had anything to logs or am I reading something wrong.

Well normally, to get the error, you would just add the relative errors. I showed you how to do it.

So if you had A=r3 then dA/A = 3*dr/r

It comes out the same if you just differentiate it normally.

I know at the beginning I asked how to use sigma and partial derivatives to solve this type of problem but I don't really know much about them yet. We haven't gotten to them in my math class. This problem is coming from an intro to physics lab course that focuses on propagation of error and uncertainty in measurements made and then using Excel functions like STDEV and (chi^2) to figure out stuff related to uncertainties.

Is there a standard formula to use if given a measurement or multiple measurements and the uncertainity in them?

"dA/A", is that supposed to be a partial derivative?

zero13428 said:
I know at the beginning I asked how to use sigma and partial derivatives to solve this type of problem but I don't really know much about them yet. We haven't gotten to them in my math class. This problem is coming from an intro to physics lab course that focuses on propagation of error and uncertainty in measurements made and then using Excel functions like STDEV and (chi^2) to figure out stuff related to uncertainties.

Is there a standard formula to use if given a measurement or multiple measurements and the uncertainity in them?

In that case, you can just find the areas with the radii given and then find the standard deviation, which would be how much the measurement deviates from the mean. Measuring its error.

zero13428 said:
"dA/A", is that supposed to be a partial derivative?

If you had like one value alone and you wanted to get the error,

dA would be the error in A.
A would be the actual measurement.

The relative error in A would then be dA/A

Actually I think I got it worked out. Let me know if this looks right.

A=(∏)(r)^2
∂(A)/∂(r) = 2(∏)(r)

sigma_A=√(((∂A/∂r)^2)(sigma_r)^2))

sigma_A=√(((2∏(14.3))^2)(0.3)^2))= 26.9cm

Area = 642.4cm
Uncertainty = 26.9cm

zero13428 said:
Actually I think I got it worked out. Let me know if this looks right.

A=(∏)(r)^2
∂(A)/∂(r) = 2(∏)(r)

sigma_A=√(((∂A/∂r)^2)(sigma_r)^2))

sigma_A=√(((2∏(14.3))^2)(0.3)^2))= 26.9cm

Area = 642.4cm
Uncertainty = 26.9cm

If you wanted to use the partial derivative ∂A, as the error in A, then it should read like this

∂A/∂r= 2πr or ∂A=2πr ∂r

Now if we divide both sides by A (which is πr2 as well)

∂A/A = 2πr/πr2 ∂r

or ∂A/A = 2∂r/r

1. What is the formula for calculating the area of a circle and how is uncertainty incorporated?

The formula for calculating the area of a circle is A = πr^2, where r is the radius of the circle. To incorporate uncertainty, we can use the formula for uncertainty propagation, which is ΔA = 2πrΔr. This means that the uncertainty in the area is equal to twice the uncertainty in the radius multiplied by π.

2. How is uncertainty in the radius of a circle determined?

The uncertainty in the radius of a circle can be determined through various methods, such as measurement error, rounding error, or instrument error. It is important to carefully consider and calculate all sources of uncertainty to get an accurate estimate of the radius.

3. Can the uncertainty in the area of a circle be greater than the actual area?

Yes, in some cases, the uncertainty in the area of a circle can be greater than the actual area. This can happen when the uncertainty in the radius is relatively large compared to the radius itself. It is important to consider all sources of uncertainty and make sure they are as small as possible to minimize the difference between the uncertainty and the actual area.

4. Is there a way to reduce uncertainty in the area of a circle?

Yes, there are several ways to reduce uncertainty in the area of a circle. One way is to increase the precision of measurements used to calculate the radius, such as using more precise instruments or taking multiple measurements and averaging them. Another way is to minimize sources of error, such as ensuring the circle is perfectly round and the measurements are taken at the center of the circle.

5. How does uncertainty in the area of a circle affect other calculations that use the area?

The uncertainty in the area of a circle can affect other calculations that use the area, such as finding the circumference or diameter. This is because the uncertainty in the area is propagated to these calculations and can lead to a larger uncertainty in the final result. It is important to consider and account for uncertainty in all calculations involving the area of a circle to ensure accurate and reliable results.

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