Quiz on Converting Between Radians and Degrees

Please give the degree measure angle equivalent to $\pi$  radians. Your answer will be an integer number of degrees, for example "75".

Answer explanation

An angle measuring around the circle is  $2\pi$  radians, or 360 degrees.  Half that is  $\pi$  radians or 180 degrees, which is your answer.  The easiest way to find the answer is probably mental math, rather than using the proportion  $\frac{\pi}{180}=\frac{\theta R}{\theta\degree}$  .

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CCSS.HSF.TF.A.1

Please give the radian measure angle equivalent to  $90\degree$ .

•  $\frac{\pi}{2}$  radians, or about 1.57 radians.

•  $\frac{\pi}{4}$  radians, or about 0.79 radians.

•  $2\pi$  radians, or about 6.28 radians.

•  $\frac{3\pi}{4}$  radians, or about 2.36 radians.

Answer explanation

 $90\degree$  is one-fourth of the degree-angle measuring around the whole circle ( $360\degree$  ), so  $\frac{\pi}{2}$  radians is the answer because it is one-fourth of the radian-angle measuring around the whole circle ( $2\pi$  ).  Again, mental math is probably easier than using the proportion  $\frac{\pi}{180}=\frac{\theta R}{\theta\degree}$   .

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CCSS.HSF.TF.A.1

Find the degree measure angle equivalent to $\frac{3\pi}{4}$ 

radians.  Your answer will be an integer number of degrees, for example, "105".

Answer explanation

Since $\frac{\pi}{4}$  radians is  $45\degree$  , three times that is your answer,  $135\degree$ .  You can also use the proportion,  $\frac{\pi}{180}=\frac{\left(\frac{3\pi}{4}\right)}{\theta\degree}$  and solve for  $\theta\degree$  .   $\theta\degree=\frac{\frac{3\pi}{4}\cdot180}{\pi}$  or $135\degree$ .

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CCSS.HSF.TF.A.1

What is the radian-measure angle equivalent to $210\degree$ 

?

•  $\frac{7\pi}{6}$  radians, or about 3.67 radians.

•  $\frac{5\pi}{6}$  radians, or about 2.62 radians.

•  $\frac{4\pi}{3}$  radians, or about 4.19 radians.

•  $\frac{2\pi}{3}$  radians, or about 2.09 radians.

Answer explanation

Mental math style: $210\degree\ =180\degree+30\degree$  , and  $180\degree=\pi$  radians,  $30\degree=\frac{\pi}{6}$  radians, so  $210\degree=\left(\pi+\frac{\pi}{6}\right)=\frac{7\pi}{6}$  radians.  Using the proportion  $\frac{\pi}{180}=\frac{\theta R}{210\degree}$   ,  $\theta R=\frac{210\cdot\pi}{180}=\frac{7\pi}{6}$  radians.

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CCSS.HSF.TF.A.1

Find the degree-measure angle equivalent to $\frac{2\pi}{3}$  radians.  Your answer will be an integer number of degrees, for example "40".

Answer explanation

 $\frac{\pi}{3}=60\degree$  , so  $\frac{2\pi}{3}=120\degree$  .  Using the proportion  $\frac{\pi}{180}=\frac{\frac{2\pi}{3}}{\theta\degree}$  ,  $\theta\degree=\frac{\frac{2\pi}{3}\cdot180}{\pi}=120\degree$  

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CCSS.HSF.TF.A.1

What is the radian-measure angle equivalent to $420\degree$  ?

•  $\frac{7\pi}{3}$  radians, or about 7.33 radians.

•  $\frac{7\pi}{6}$  radians, or about 3.67 radians.

•  $\frac{6\pi}{7}$  radians, or about 2.69 radians.

•  $\frac{3\pi}{7}$  radians, or about 1.35 radians.

Answer explanation

 $\frac{\pi}{180}=\frac{\theta R}{420\degree}$  , so  $\theta R=\frac{420\pi}{180}=\frac{7\pi}{3}$  .  This angle is bigger than  $360\degree$  , so its equivalent is bigger than  $2\pi$   radians.

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CCSS.HSF.TF.A.1

What degree-measure angle is equivalent to $-\frac{\pi}{6}$  radians?  (Note this time, the angle is negative.)  Give an integer number of degrees, for example "175".

Answer explanation

 $\frac{\pi}{6}=30\degree$  , so  $-\frac{\pi}{6}=-30\degree$  .  Using the proportion,  $\frac{\pi}{180}=\frac{-\frac{\pi}{6}}{\theta\degree}$  , so  $\theta\degree=\frac{-\frac{\pi}{6}\cdot180}{\pi}=-30\degree$  .

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CCSS.HSF.TF.A.1