If I want to transform from degrees to radians, I have the right to use the role that takes level value as an input, multiplies it v $fracpi180$ and returns the radian value: $operatornameDtoR(d)=d imes fracpi180$.

You are watching: Pi/18 on unit circle

And if I desire to walk from radians to degrees I must only go backwards and also divide radian value through $fracpi180$ (e.g. Multiply it v $frac180pi$): $operatornameRtoD(r)=r imes frac180pi$.

My inquiry is this: Why does multiplying/dividing with $fracpi180$ converts levels into radians/radians right into degrees? Why precisely that number, not some other? Also, does this job-related only because that unit circle, or for any kind of circle?



A full circle is $360^circ$ and also $2pi$ radians. Hence $360^circ = 2pi ext rad$. We simplify by separating by two.


Take a look at at this ratios:$frac180d=fracpir$

Where $d$ is the degrees and $r$ is radians.Knowing the $pi$ radians is 180 degrees one deserve to setup this proportion to uncover the values they"re spring for.

Finding degrees: $d=frac180pir$Finding radians: $r=fracpi180d$

These equations room simply acquired from the very first ratio.

It doesn"t only must be $fracpi180$, that can additionally be setup as:$frac360d=frac2pir$Because the is likewise known that $2pi$ radians is a full change about the circle simply as 360 degrees is.


Think of the as solving proportions. We have $$pi ext radians = 180^circ$$ and also you desire to convert $$r ext radians = d^circ.$$ hence you get$$fracpir = frac180d$$which simplifies to$$r = fracpi180 d quad extor equivalently quad d = frac180pi r.$$


The radian is defined as the airplane angle subtended by any type of circular arc divided by its radius.

When the one arc is actually congruent come the circle, the size is $2pi r=2pi=$ $ au$ (for a unit circle). The edge subtended by this arc is $360^ exto$, and also therefore $1: extradian=frac360 au=frac180pi$.

So: $$r ext radians= extdcdot frac180pi\ d ext degrees = fracrfrac180pi=frac180rpi$$

Since circumference =2pier if we take fifty percent circle it would be pier in i beg your pardon rotation of edge is 180 degree and also arc size would be pie*r climate we acquire theta =pie =180 degree

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