# find arclength given angle of circle in degrees and radius oif circle

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

$$\frac{80}{360}=\frac{2}{9}$$

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

• $\cfrac{80}{360} =\cfrac{2}{9}$... and $\cfrac{2 \times 12 \pi}{9}=\cfrac{24 \pi}{9}$ Nov 28, 2014 at 23:53
• My only guess is that the answer should be $\cfrac{24 \pi}{9}$ and your book approximated it to 2 significant digits. Nov 28, 2014 at 23:58
• ok so my question is how do we convert 24/9 pi radians into inches
Nov 28, 2014 at 23:59
• 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 $\frac{24}{9} \pi = 8.37758\dots$.
– Ian
Nov 29, 2014 at 0:00
• so I just use 3.14 for pi basically? oh jeeze.. smh. im sorry. Thanks for the help though :D
Nov 29, 2014 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: $$\frac{80}{360}=\frac{2}{9} \rightarrow \frac{2}{9}\times (2r\pi)=\frac{2}{9}\times (12\pi)=\frac{24\pi}{9}$$