Angle Measures in Degrees and Radians

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Pre-Calculus › Angle Measures in Degrees and Radians

Questions 1 - 10

Determine the value of $225^\circ$ in radians.

$\frac{5\pi}{4}rad$

$\frac{40500}{\pi}rad$

$\frac{3\pi}{4}rad$

$\frac{5\pi}{3}rad$

Explanation

To convert from degrees to radians, you do:

$225deg\cdot \frac{(\pi) rad}{180 deg}= \frac{(45\cdot5)\cdot\pi}{45\cdot4} rad= \frac{5\pi}{4} rad$

Find the length of an arc of a circle if the radius is and the angle is $\frac{\pi}{4}$ radians.

$\frac{3\pi}{4}$

$\frac{5\pi}{4}$

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

$\frac{\pi}{4}$

$\frac{\pi}{2}$

Explanation

Write the formula to find the arc length given the angle in radians.

$s=r\theta$

Substitute the radius and angle.

$s=3\left(\frac{\pi}{4}\right)=\frac{3\pi}{4}$

Convert $660^{\circ}$ to radians.

$\frac{11\pi }{3}$

$\frac{66\pi }{7}$

$\frac{13\pi }{3}$

$\frac{8\pi }{3}$

$\frac{111\pi }{3}$

Explanation

To convert from degrees to radians, you must multiply by $\frac{\pi }{180}$ : $660\cdot \frac{\pi }{180}$ . Then, multiply across as you do normally with fractions. Try and simplify if you can. In this case, 60 goes into both 660 and 180.

Thus, your answer is $\frac{11\pi }{3}$ .

Determine the value of $225^\circ$ in radians.

$\frac{5\pi}{4}rad$

$\frac{40500}{\pi}rad$

$\frac{3\pi}{4}rad$

$\frac{5\pi}{3}rad$

Explanation

To convert from degrees to radians, you do:

$225deg\cdot \frac{(\pi) rad}{180 deg}= \frac{(45\cdot5)\cdot\pi}{45\cdot4} rad= \frac{5\pi}{4} rad$

Find the length of an arc of a circle if the radius is and the angle is $\frac{\pi}{4}$ radians.

$\frac{3\pi}{4}$

$\frac{5\pi}{4}$

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

$\frac{\pi}{4}$

$\frac{\pi}{2}$

Explanation

Write the formula to find the arc length given the angle in radians.

$s=r\theta$

Substitute the radius and angle.

$s=3\left(\frac{\pi}{4}\right)=\frac{3\pi}{4}$

Convert $660^{\circ}$ to radians.

$\frac{11\pi }{3}$

$\frac{66\pi }{7}$

$\frac{13\pi }{3}$

$\frac{8\pi }{3}$

$\frac{111\pi }{3}$

Explanation

Thus, your answer is $\frac{11\pi }{3}$ .

Find the area of a sector with the radius of 1 and angle of $\frac{\pi}{3}$ .

$\frac{\pi}{6}$

$\frac{\pi}{3}$

$\frac{\pi}{4}$

$\frac{\pi}{2}$

$\pi$

Explanation

Write the formula for the area of a sector in radians.

$A=\frac{1}{2}r^2\theta$

Substitute the radius and the angle.

$A=\frac{1}{2}(1)^2(\frac{\pi}{3})=\frac{\pi}{6}$

Find the arc length of a circle with circumference $10\pi$ that goes from to $135^\circ$ around its center.

$A=\frac{15\pi}{4}$

$A=\frac{25\pi}{4}$

$A=\frac{10\pi}{3}$

Explanation

Remember that arc length of a circle is given by:

$A=\theta r$ , where $\theta$ is angle made by connecting the radius to the two ends of the arc, given in radians. is the radius.

To get $\theta$ , we convert from degrees to radians.

$\theta = 135deg\cdot\frac{\pi rad}{180deg}=\frac{3\pi}{4}$

To get the radius we convert from the circumference

$r=\frac{C}{2\pi}=\frac{10\pi}{2\pi}=5$

$A=\frac{3\pi}{4}\cdot5= \frac{15\pi}{4}$

If circle has a radius of ${}\frac{5}{\pi}$ fathoms and sector has a central angle of $\pi$ . What is the measure of arc ?

$5\pi fathoms$

$\frac{5}{\pi} fathoms$

Explanation

To find the length of an arc, use the following formula:

$\frac{\theta }{2\pi}\cdot 2\pi r=arc_{length}$

We are given the radius, as well as our central angle, so plug in what we know and simplify

$\frac{\pi }{2\pi}\cdot 2\pi \cdot \frac{5}{\pi}=\frac{1}{2}\cdot 2\cdot 10=5$

So our answer is 5 fathoms

If circle has a radius of ${}\frac{5}{\pi}$ fathoms and sector has a central angle of $\pi$ . What is the measure of arc ?

$5\pi fathoms$

$\frac{5}{\pi} fathoms$

Explanation

To find the length of an arc, use the following formula:

$\frac{\theta }{2\pi}\cdot 2\pi r=arc_{length}$

We are given the radius, as well as our central angle, so plug in what we know and simplify

$\frac{\pi }{2\pi}\cdot 2\pi \cdot \frac{5}{\pi}=\frac{1}{2}\cdot 2\cdot 10=5$

So our answer is 5 fathoms

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