# What is the area, and approximate uncertainty in a circle....

• JustynSC
In summary, the conversation is discussing the calculation of the area and uncertainty of a circle with a given radius of 3.1*10^4 cm. The equation for the area of a circle is used to solve for the area, with the resulting answer being 3.0+/-0.1e9 cm^2. However, there is confusion about the uncertainty of 0.2e9 cm^2 in the book's answer and how it was derived. The book's note states that numbers are accurate to +/- 0.1, which may explain the difference in uncertainty.
JustynSC

## Homework Statement

What is the area, and approximate uncertainty in a circle with radius 3.1*10^4 cm (or written: 3.1e4 cm)?

Area=Pi*r^2

## The Attempt at a Solution

My attempt to the solution took some trial and error, and it went as follows:
Substitute the circle's radius into the equation for the area of the circle: A=Pi(3.1e4)^2
Then I squared the () : A=Pi(9.61e8)
Following this I Multiplied by Pi: 3.017e9 cm^2 (sig fig) ==> 3.0e9 cm^2

This answer above Is correct, but in the book the answer is 3.0+/-0.2e9 cm^2.
The part that I do not understand is that the uncertainty they predict is +/-0.2

I figured that the uncertainty should be +/- 0.1e4, giving the radius a minimum of 3.0e4, and a max of 3.2e4.
Based on my way of working though the problem, my answer come out to be A=3.0+/-0.1e9 cm^2

If anyone can explain why my uncertainty isn't correct, and how they get that answer that would be wonderful!

I agree with your answer, but the error in 3.1 is +/-0.05, not +/-0.1. That's a 1 in 60 error, so the error in the area should be 1 in 30.

haruspex said:
I agree with your answer, but the error in 3.1 is +/-0.05, not +/-0.1. That's a 1 in 60 error, so the error in the area should be 1 in 30.
How do you come up with the error as 0.05 when the device used to measure does not specify that is has the exact measurements to the hundredth of a cm? In other words, if the .00 is not represented, how can it be used as the error?
Thirdly, I do not understand what you mean by 1 in 60 and 1 in 30

JustynSC said:
How do you come up with the error as 0.05 when the device used to measure does not specify that is has the exact measurements to the hundredth of a cm? In other words, if the .00 is not represented, how can it be used as the error?
Thirdly, I do not understand what you mean by 1 in 60 and 1 in 30
You did not specify an error range for the radius, so it is implied by the number of significant digits. Quoting 3.1 implies something from 3.05 to 3.15. Had the radius been between 3.00 and 3.05?then the stated measurement should have been 3.0.

An error of 0.05 in a measurement of about 3 is a one part in sixty error, 0.05/3=1/60.

Thanks for the help! I get it now.

haruspex said:
You did not specify an error range for the radius, so it is implied by the number of significant digits. Quoting 3.1 implies something from 3.05 to 3.15. Had the radius been between 3.00 and 3.05?then the stated measurement should have been 3.0.

An error of 0.05 in a measurement of about 3 is a one part in sixty error, 0.05/3=1/60.
So I was looking into the problem a little further and I might have missed a minor detail that explains why they did not use +/- 0.05 as the error. But I still can't figure out how they justify the answer to be within +/- 0.2
Here is the Note at the start of all problems:
assume a number like 6.4 is accurate to +/- 0.01, and 950 is +/-10 unless 950 is said to be "precisely" or "very nearly" 950, in which case assume 950 +/- 1.
I hope this help explain how they have their answer in the back of the book differing from ours.

JustynSC said:
So I was looking into the problem a little further and I might have missed a minor detail that explains why they did not use +/- 0.05 as the error. But I still can't figure out how they justify the answer to be within +/- 0.2
Here is the Note at the start of all problems:
assume a number like 6.4 is accurate to +/- 0.01, and 950 is +/-10 unless 950 is said to be "precisely" or "very nearly" 950, in which case assume 950 +/- 1.
I hope this help explain how they have their answer in the back of the book differing from ours.
I believe you have a typo, and that the book says - or should have said - something like
assume a number like 6.4 is accurate to ± 0.1
not ± 0.01.

JustynSC said:
So I was looking into the problem a little further and I might have missed a minor detail that explains why they did not use +/- 0.05 as the error. But I still can't figure out how they justify the answer to be within +/- 0.2
Here is the Note at the start of all problems:
assume a number like 6.4 is accurate to +/- 0.01, and 950 is +/-10 unless 950 is said to be "precisely" or "very nearly" 950, in which case assume 950 +/- 1.
I hope this help explain how they have their answer in the back of the book differing from ours.
Assuming, as SammyS says, that should read 6.4+/-0.1...
Ok, but that is a bit unusual. One would normally take 6.4 as being accurate to that many figures, so represents a range 6.35 to 6.45.
JustynSC said:
giving the radius a minimum of 3.0e4, and a max of 3.2e4
Ok, but what areas do you calculate from those two radii?

## What is the area of a circle?

The area of a circle is the amount of space enclosed within its boundaries. It is calculated by multiplying the square of the radius (the distance from the center of the circle to the edge) by pi (approximately 3.14).

## How do you find the area of a circle?

The formula for finding the area of a circle is A = πr², where A is the area and r is the radius. Simply plug in the value of the radius into the formula and solve for the area.

## What is the approximate uncertainty in a circle's area?

The uncertainty in a circle's area depends on the uncertainty in its radius. Generally, the uncertainty in area will be twice the uncertainty in the radius. For example, if the radius has an uncertainty of 0.5 units, the area will have an uncertainty of approximately 1 square unit.

## Can the area of a circle be negative?

No, the area of a circle cannot be negative. It is always a positive value, since it represents the amount of space enclosed within the circle's boundaries.

## How accurate is the approximation of pi (π) in calculating the area of a circle?

The approximation of pi used in calculating the area of a circle (3.14) is accurate to two decimal places. However, for more precise calculations, a more accurate value of pi can be used, such as 3.14159.

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