# Finding The Number Of Degrees In An Angle Using Radian Theorem.

Is this correct? The question is.

Find the number of degrees subtended at the center of a circle by an arc whose length is 0.357 time the radius, taking $\pi = 3.1416$

To help solve this, one very common trigonometry theorem explains that

$\angle AOP\over A\space Radian$ = $arc AP\over Radius$

Since $A\space Radian$ $=$ $180\over \pi$ $= Radius$... we may multiply $180\over \pi$ $\times 0.357$ to obtain an arc length of 57.29 units in length.

Dividing the arc length by 0.357 will yield the radius length . So $57.29 \over 0.357$ $=$ $160.47 units$

Lastly, since $\angle AOP$ $=$ $arcAP \over Radius$ $\times$ $A \space Radian$, then $\angle AOP$ =$57.29 \over 160$ $\times$ $180\over\pi$ =$20.515\space degrees \space nearly$

The calculations in the posted question show that indeed sometimes two (or more) wrongs do make a right.

But first, let's go over what you did right, and finish it correctly. From $\frac{\angle AOP}{1\text{ radian}} = \frac{\text{arc }AP}{\text{radius}}$ you correctly deduce (on the last line of the posted question) that

$$\angle AOP = \frac{\text{arc }AP}{\text{radius}} \times (1\text{ radian}).$$

You want to apply this formula to "an arc whose length is $0.357$ times the radius". To do this, let the arc $AP$ to be the length of arc described in the quoted text. In other words, let

$$\text{arc }AP = 0.357 \times \text{radius}.$$

Now plug this value in for $\text{arc }AP$ in the previous equation:

$$\angle AOP = \frac{0.357 \times \text{radius}}{\text{radius}} \times (1\text{ radian}).$$

But $\frac{\text{radius}}{\text{radius}} = 1$, so

$$\angle AOP = 0.357 \times (1\text{ radian}).$$

To find the number of degrees in this angle, you can use the fact that $1\text{ radian} = \frac{180}{\pi}\text{ degrees}$, so

$$\angle AOP = 0.357 \times \left(\frac{180}{\pi}\text{ degrees}\right) \approx 20.4545 \text{ degrees}.$$

Miraculously, the number you came up with is almost the same as this; the difference is due to roundoff in several places in your calculations, mostly when you rounded $160.47$ to $160$.

Now for some things not to do.

When you write "${180\over \pi} = \text{Radius}$" you are asserting that every circle has a radius of ${180\over \pi}$. This is surely false. There are circles of radius $1$ and circles of radius $1000.567$, to name just a couple of counterexamples.

In other words, there is no justification to say the radius of your circle is ${180\over \pi}$. Therefore there is no justification to say that you "multiply ${180\over \pi}\times 0.357$" to obtain the arc length.

Moreover, ${180\over \pi}\times 0.357$ is not even close to $57.29$. As we already saw, ${180\over \pi}\times 0.357 \approx 20.4545$. Apparently what happened was that in considering ${180\over \pi}\times 0.357$, you did the division and neglected the multiplication:

$${180\over \pi} \approx 57.2956.$$

That is, you merely computed the conversion factor from radians to degrees: an angle of one radian is an angle of approximately $57.2956$ degrees.

But now, having stated that the length of the arc is $57.2956$, you start to consider a different circle. Assuming that $57.2956$ is the length of the desired arc on a circle of suitable radius, from this point forward you calculate correctly: indeed the radius of that circle must be approximately $160.47$, and you can plug that value into the formula $\angle AOP = \frac{\text{arc }AP}{\text{radius}} \times (1\text{ radian})$ to compute the angle of the arc.

The reason this worked as well as it did was that the answer is independent of the radius of the circle, and therefore you can guess any radius you like and still come out with the correct angle measure, as long as the number you use for the length of the arc is $0.357$ times the number you use for the radius.

What I would worry about if I were you is how not to make the wrong steps that you took in the process of getting the answer. If I were grading this as a homework exercise I would probably give partial credit for the bit at the end, after you plugged in $160$ as the radius, but I would take off most of the point value of the question due to the multiple errors that led you to assert that the radius must be $160$.

• This was quite an eye oponer. It looks like the LHS contain values independent of a circle's simagnitude, while the RHS does not. This must mean that the same logic that you told me about for the denominator, must apply with the numerator as well. Thanks! – Lump Coon Nov 12 '15 at 1:54

Last two lines seem meaningless. You arrived at your answer in the third line itself.

$$\angle AOP = \frac{\text{arc AP}}{\text{radius}} \times 1^c$$ $$= 0.357 \times \frac{180}{\pi}$$ $$\approx 20.45^\circ$$