I'm having a lot of difficulty with getting this to make sense and the answer in the book is just '8.4 in'

Q " You want to make an 80 degree angle by marking an arc oin the perimeter of a 12-in. diameter disk and drawing lines from the ends of the arc to the disk's center. To the nearest tenth of an inch, how long should the arc be?"

I tried going about it 2 different ways...

I converted 80 degrees into radians in order to use the special property that theta in radians = arclength/radius or: $$ \frac{80}{180}\pi = \frac{4}{9}\pi $$

And by $$\theta=\frac{s}{r},$$ $$\frac{4}{9}\pi=\frac{s}{6}$$ So, $$s = \frac{24}{9}\pi $$ But why?! Why then does my book say 8.4 inches!

So then I tried going about it with my stupid head instead of the book... and I thought OK well a circle is 360 degrees.. lets see if I can work it out that way


So 2/9's of the distance around the circle which I know to be:$$2r\pi = 12\pi$$ inches is:$$\frac{24}{9}\pi = \frac{24}{9}*3.14 = ~~8.37$$

  • $\begingroup$ $\cfrac{80}{360} =\cfrac{2}{9}$... and $\cfrac{2 \times 12 \pi}{9}=\cfrac{24 \pi}{9}$ $\endgroup$ – user160738 Nov 28 '14 at 23:53
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    $\begingroup$ My only guess is that the answer should be $\cfrac{24 \pi}{9}$ and your book approximated it to 2 significant digits. $\endgroup$ – user160738 Nov 28 '14 at 23:58
  • $\begingroup$ ok so my question is how do we convert 24/9 pi radians into inches $\endgroup$ – Adam Nov 28 '14 at 23:59
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    $\begingroup$ It's not $\frac{24}{9} \pi$ radians. It's $\frac{24}{9} \pi$ inches: you multiplied radians (which can be thought of as dimensionless for this purpose) times inches, so you got inches. Then $\frac{24}{9} \pi = 8.37758\dots$. $\endgroup$ – Ian Nov 29 '14 at 0:00
  • $\begingroup$ so I just use 3.14 for pi basically? oh jeeze.. smh. im sorry. Thanks for the help though :D $\endgroup$ – Adam Nov 29 '14 at 0:01

$$\frac{24}{9}\pi\; \text{ inches} \approx 8.3775804097\;\text{inches} \approx 8.4 \;\text{inches} $$ And for your second method it should give the same as your first: $$\frac{80}{360}=\frac{2}{9} \rightarrow \frac{2}{9}\times (2r\pi)=\frac{2}{9}\times (12\pi)=\frac{24\pi}{9}$$

  • $\begingroup$ yeah but my question was how? and where did I go wrong ending up with 4 pi? $\endgroup$ – Adam Nov 28 '14 at 23:51
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    $\begingroup$ Do you mean why don't the two methods give the same thing? $\endgroup$ – Jay Nov 28 '14 at 23:52
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    $\begingroup$ @RossMillikan Cheers! didn't spot it. $\endgroup$ – Jay Nov 29 '14 at 0:06

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