Need some clarification regarding radians

I was sitting the other day with my cousin, helping him with his homework, when he asked me what's the deal with these radians.

So I took a paper and a pen and tried to recall where did these radians come from, but something is not quite clear for me.

Consider the following circle:

http://i41.tinypic.com/1trhg5.png

When the angle is 90 degrees, arc **a** is exactly $\displaystyle \frac{1}{4}$ the length of the full circle's perimeter.

and therfor the angle in radians is:

$\displaystyle \frac{1}{4}\cdot\2\pi r$

So when r=1, we're good, we get: $\displaystyle \frac{1}{4}\cdot\2\pi = \frac{\pi}{2}=90^{\circ}$

But obviously, it is not the case for any other r, as it should be...

So what am I missing here?

Thanks in advanced!

Re: Need some clarification regarding radians

Quote:

Originally Posted by

**Stormey** I was sitting the other day with my cousin, helping him with his homework, when he asked me what's the deal with these radians.

So I took a paper and a pen and tried to recall where did these radians come from, but something is not quite clear for me.

Consider the following circle:

http://i41.tinypic.com/1trhg5.png
When the angle is 90 degrees, arc

**a** is exactly $\displaystyle \frac{1}{4}$ the length of the full circle's perimeter.

and therfor the angle in radians is:

$\displaystyle \frac{1}{4}\cdot\2\pi r$

So when r=1, we're good, we get: $\displaystyle \frac{1}{4}\cdot\2\pi = \frac{\pi}{2}=90^{\circ}$

But obviously, it is not the case for any other r, as it should be...

So what am I missing here?

What you are missing is the fact that **the "size" measure of the angle does not change with a change in radius**.

Now the length of the arc changes but not the measure of the angle.

Re: Need some clarification regarding radians

Hi Plato, thanks for your post.

The property you mentioned (that the change in radius doesn't affect the angle) is understood (it's quite intuitive actually), but still, we need to somehow **define** radian.

Do we arbitrarily define the radius to be 1, when we talk about radians?

Re: Need some clarification regarding radians

Quote:

Originally Posted by

**Stormey** **define** radian.

Do we arbitrarily define the radius to be 1, when we talk about radians?

Next time please state the question that you mean to ask clearly as you did above.

**One radian is the measure of the central angle in a circle that subtends an arc of length equal to the radius of the circle.**

Re: Need some clarification regarding radians

Thanks.

and btw, I'm not an English speaker, so sometimes it's not so easy to phrase the question properly.

can you tell me what was wrong with it?

Re: Need some clarification regarding radians

Hi Stormey. Ther error is here:

Quote:

Originally Posted by

**Stormey** When the angle is 90 degrees, arc **a** is exactly $\displaystyle \frac{1}{4}$ the length of the full circle's perimeter.

and therfor the angle in radians is:

$\displaystyle \frac{1}{4}\cdot\2\pi r$

Given the definition of the radian as Plato indicated, this should be:

"When the angle is 90 degrees, arc **a** is exactly $\displaystyle \frac{1}{4}$ the length of the full circle's perimeter.

and therfore the angle in radians is:

angle = $\displaystyle \frac {arc \ length} {Radius} = \frac {( \frac{1}{4}\cdot\2\pi r)}{r} = \frac {\pi} 2$

Re: Need some clarification regarding radians

Equivalently, the radian measure of an angle is the length of the arc of circle the angle subtends **divided by** the radius of the circle. Of course, if the radius is 1, that reduces to "the length of the arc". But if the circle has radius "R" then it has circumference $\displaystyle 2\pi R$. One fourth or the circle has circumference $\displaystyle \frac{2\pi R}{4}= \frac{\pi R}{2}$ so that the radian measure of this 90 degree angle is $\displaystyle \frac{\pi R}{2R}= \frac{\pi}{R}$.

Re: Need some clarification regarding radians

When you are given a radian measure it literally means "how many lengths of the radius lie on the circumference when the angle has been swept out?"