**Introduction**

We have learned about angle measures since elementary grades. In Figure 1, we have a circle with center ** A**, and radius length

**1**. Angle

**measures 90 degrees and**

*CAB**intercepting*minor arc

**. This is also the same as saying that arc**

*BC***is**

*BC**subtending*angle

**. We have also learned that the entire rotation about the center of a circle is 360**

*CAB**degrees*.

Another unit of angle measure besides degree is radian. Now, what is radian? How is it related to degree?

Consider the second figure shown below. Segment ** OP** is the radius of the circle with center

**, and**

*O***is the arc subtending angle**

*NP***. If**

*NOP***is a string that wraps the portion of a circle, if we “straightened” it and its length equals the length of**

*NP***, then we say that angle**

*OP***is equal to one radian. Therefore,**

*NOP***One radian is equal to the angle measure at the center of a circle subtended by an arc equal in length to the radius.**

If the length of ** NP = r**, then how many radians is the entire rotation of point

**about the center**

*P***?**

*O*We know that the circumference of the circle equals $katex 2 \pi r$. Note that angle** NOP, **which intercepts arc

**equals**

*NP*,**1 radian**. Now, we are looking for

**, which intercepts the**

*x***circumference of the circle**. Thus, we have the following ratio

**: angle NOP : radius length = x : circumference of the circle.**But angle

**is equal to 1 radian and**

*NOP***the circumference of the circle is equal to . Substituting we have**

Now, the ratio above is the same as . Solving for , we have . This means that the entire rotation about the circle is equal to radians.

We have mentioned that the rotation about the center of a circle is 360 degrees. Therefore, radians = degrees. or equivalently, radians degrees. These are the only equations that we should remember especially when converting degree to radian measure and vice versa.

**Conversion**

To give us a feel of the worth of our discussion above, let us have a few examples on degree-radian conversions. We will use ratio and proportion to convert degrees to radian and vice versa.

*Example 1***:** Convert to radians.

Solution: We go back to what we know: radians **= ** degrees**. **Now, let’s use ratio and proportion: . It follows that . Dividing both sides by 180, we have . Hence, 120 degrees equals radians.

* *

*Example 2***:** Convert to degrees.

*Solution:* Let* y* be the measure of in radians. Again, using ratio and proportion, we have . Solving, we have degrees.

As you can see, you don’t really have to memorize which to multiply, or in conversion.

**Generalization**

Of course, we can generalize the situation above. For instance where is the given degree measure. Solving we have . That’s the reason why you multiply the degree measure by when converting degree to radian measure. But as I have said earlier, no need to memorize. Just use ratio and proportion.

**Exercise**: Show the derivation of converting radian to degree measure.

*Multiplying the mean, which are the two inner numbers and the extremes, which are the two outer numbers. This is valid because means, . Therefore, .