Understanding Radians and Conversions - Math

Card 0 of 188

Question

Bob manages a pizza store. He bought a new machine that tracks how big his employees are cutting the pizza slices. The machine measures the average angle size of each slice of each pizza. Unfortunately, the angle is given as 0.7854 radians which Bob does not understand. Help Bob by converting the radian angle into degrees. In degrees, what is the size of the angle for an average pizza slice.

Answer

To convert we use a common conversion amount. It may be easiest to remember the full circle example. In degrees, a full circle is $360^{\circ}$ around. In terms of radians, a full circle is $2\pi$ . So to get our answer

$0.7854\ \text{Radians} *\frac{360^{\circ}}{2\pi\ \text{Radians}}= 45^{\circ}$

Compare your answer with the correct one above

Question

Convert $120^{\circ}$ into radians.

Answer

To convert from degrees to radians, one multiplies by $\frac{\pi }{180}$ .

$120\cdot \frac{\pi }{180}=\frac{2\pi}{3}$

Compare your answer with the correct one above

Question

What is the value of the angle $\frac{3\pi}{2}$ radians when converted to degrees?

Answer

The answer can be found using the conversion of 1 radian equals $\frac{180}{\pi}$ degrees. Multiplying $\frac{3\pi}{2}$ by this conversion factor gives 270 degrees.

Compare your answer with the correct one above

Question

Express in radians: $125^{\circ }$

Answer

Since $180 ^{\circ } = \pi \textrm{ rad}$ , we can convert as follows:

$125 \cdot \frac{\pi }{180} = \frac{25\pi }{36}$

Compare your answer with the correct one above

Question

A point has Cartesian coordinates . Rewrite this as an ordered pair in the polar coordinate plane, rounding the coordinates to the nearest hundredth.

Answer

Set . Calculate the polar coordinates $(r,\theta )$ as follows:

$r = \sqrt{x^{2}+y^{2}}= \sqrt{1^{2}+3^{2}}=\sqrt{10} \approx 3.16$

$\theta = \arctan \frac{y}{x}= \arctan \frac{3}{1} = \arctan 3 \approx 1.25$

Compare your answer with the correct one above

Question

How many radians are in $270^\circ$ ?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{270^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$270^\circ\pi=180^\circx$

Isolate :

$\frac{270^\circ}{180^\circ}\pi=x$

$\frac{3\pi}{2}=x$

Compare your answer with the correct one above

Question

How many degrees are in $\frac{\pi}{3}$ radians?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{\pi}{3}$

Divide both sides by $\pi$ :

$x^\circ=\frac{180^\circ}{3}$

$x^\circ=60^\circ$

Compare your answer with the correct one above

Question

How many degrees are in $\frac{8\pi}{3}$ radians?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{8\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{8\pi}{3}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{8}{3}$

$x^\circ=480^\circ$

Compare your answer with the correct one above

Question

How many degrees are in $\frac{3\pi}{4}$ radians?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{3\pi}{4}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{3\pi}{4}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{3}{4}$

$x^\circ=135^\circ$

Compare your answer with the correct one above

Question

How many degrees are in $\frac{2\pi}{3}$ radians?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{2\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{2\pi}{3}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{2}{3}$

$x^\circ=120^\circ$

Compare your answer with the correct one above

Question

How many degrees are in $\frac{5\pi}{6}$ radians?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{5\pi}{6}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{5\pi}{6}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{5}{6}$

$x^\circ=150^\circ$

Compare your answer with the correct one above

Question

How many degrees are in $\frac{\pi}{6}$ radians?

Answer

$\frac{x^\circ}{\frac{\pi}{6}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{\pi}{6}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{1}{6}$

$x^\circ=30^\circ$

Compare your answer with the correct one above

Question

How many radians are in $120^\circ$ ?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{120^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$120^\circ\pi=180^\circx$

Isolate :

$\frac{120^\circ}{180^\circ}\pi=x$

$\frac{2\pi}{3}=x$

Compare your answer with the correct one above

Question

How many radians are in $360^\circ$ ?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{360^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$360^\circ\pi=180^\circx$

Isolate :

$\frac{360^\circ}{180^\circ}\pi=x$

$2\pi=x$

Compare your answer with the correct one above

Question

How many radians are in $405^\circ$ ?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{405^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$405^\circ\pi=180^\circx$

Isolate :

$\frac{405^\circ}{180^\circ}\pi=x$

$\frac{9\pi}{4}=x$

Compare your answer with the correct one above

Question

How many radians are in $81^\circ$ ?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{81^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$81^\circ\pi=180^\circx$

Isolate :

$\frac{81^\circ}{180^\circ}\pi=x$

Reduce:

$\frac{9\pi}{20}=x$

Compare your answer with the correct one above

Question

How many radians are in $90^\circ$ ?

Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{90^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$90^\circ\pi=180^\circx$

Isolate :

$\frac{90^\circ}{180^\circ}\pi=x$

Reduce:

$\frac{\pi}{2}=x$

Compare your answer with the correct one above

Question

How many degrees are in $2\pi$ radians?

Answer

Remember, $180^\circ=\pi$ radians, so $2\pi$ radians would correspond to $2*180^\circ=360^\circ$ .

Compare your answer with the correct one above

Question

How many radians are in $180^\circ$ ?

Answer

The relationship between degrees and radians is $180^\circ=\pi$ radians. Therefore, $180^\circ$ would be $\pi$ radians.

Compare your answer with the correct one above

Question

Convert $315^{\circ}$ to radians.

Answer

Multiply the given degrees measurement by $\frac{\pi }{180}$ .

Simplify and you get $\frac{7\pi }{4}$ .

Compare your answer with the correct one above

Tap the card to reveal the answer