# Formula for the area of a segment of a circle… What am I doing wrong!?

Alright, I'm trying to study for a big test and I know that the area of a segment of a circle will be on it. Problem is, I can't seem to get the formula for it to work. I've gone over it countless times, placed it in Google, etc. to no avail.

I'm taking a look at this site here: http://www.mathopenref.com/segmentarea.html

I've currently got the inside angle to 180 degrees so make it easier for me to make sure my method was correct.

Here are the details:

A circle with: -Radius of 3.5 -Central Angle of 180 degrees

It says to use this formula:

So, this should give me a result of about 19.2 which (because of the central angle I chose) can easily be proven by getting the entire circles area (PI*3.5^2=38.48451) and then dividing it by two (38.48451/2=19.24...). So shouldn't this work when doing it in the formula?

3.5^2/2*(PI/180*180 -(sin 180))

Problem is, I always get 24.1493149, which is incorrect. I've tried my handheld calculator, I've even tried the Google calculator with this equation.

I'm obviously making some kind of flaw in the process. Can somebody please explain what though?

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Tell the Google calculator to use degrees for sine: replace '(sin 180)' with '(sin 180 deg)'. – CiaPan Aug 19 '14 at 12:53

Your calculator is accidentally in radian mode! The sine of $180$ degrees is $0$. If you use that you will get the right answer. If angles on your test will be in degrees, make sure that your calculator is in degree mode. Mine, even if I put it in radian mode, wakes up in degree mode if it is turned off and then on. Make sure that you know how to change modes. If the calculator is in radian mode, there likely is a little R in the display.

The sine of $180$ radians is about $-0.8011526$. If I use that, I get exactly the (incorrect) $24.1493\dots$ that you keep getting.

The Google calculator also assumes that the input is in radians. In the Google link that you gave, change the "sin($180$)" to "sin($180$ degrees)". Google will then give the right answer to your problem. It is nice to see that Google people give priority to radians.

Remark: Usually errors come from a calculator that is in degree mode when it should be in radian mode. This may be the first time I have seen the reverse problem.

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Wow, I should have realized that! I had just cleared my own calculator cleared and I guess I should have gone back through to fix those things :) +1 for you and a answer check! – Xander Lamkins Jan 26 '12 at 10:39

Google evaluates the computations in radians while your number $180$ is in degrees. The same probably goes for your calculator.

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