Radians
As mentioned in Basics of Geometry Target C, angles can be measured in degrees or in radians. A radian is the measure of an angle at the center of a circle whose arc is equal in length to the radius, as seen in the applet below.
As mentioned in Basics of Geometry Target C, angles can be measured in degrees or in radians. A radian is the measure of an angle at the center of a circle whose arc is equal in length to the radius, as seen in the applet below.
From the radian investigation above we see that there are \(2\pi\) radians in a circle. Which means \(\pi\) radians for a semicircle, or \(\pi\) radians = \(180^{\circ}\).
Example 1:
What is \(135^{\circ}\) in radians?
Solution:
Converting from degrees to radians \((135^{\circ})\left(\Large\frac{\pi}{180^{\circ}}\right) = \Large\frac{135\pi}{180} = \Large\frac{45\cdot3\pi}{45\cdot4}=\Large\frac{3\pi}{4}\)
Example 2:
How many degrees is \(\Large\frac{\pi}{4}\) radians?
Solution 2:
Converting from radians to degrees \(\Large\frac{\pi}{4}\cdot\frac{180^{\circ}}{\pi}\normalsize{=}\Large\frac{180^{\circ}\pi}{4\pi}\normalsize{=}\Large\frac{45^{\circ}\cdot4\pi}{4\pi}\normalsize = 45^{\circ}\)
Another way of solving this conversion is since \(\pi = 180^{\circ}\) we can substitute \(180^{\circ}\) in for \(\pi\). So \(\Large\frac{\pi}{4}\normalsize{=}\Large\frac{180^{\circ}}{4}\normalsize{=}\Large\frac{4\cdot45^{\circ}}{4} \normalsize = 45^{\circ}\)
Arc Length and Radians
We can also find arc length when the measure of the arc is given in radians rather than degrees. We will use our conversion for radians to derive a new arc length formula. Changing the radius, looking at the circumference, and looking at the arc length (in red) we see that the arc length is a fraction of the circumference.
Example 1:
What is \(135^{\circ}\) in radians?
Solution:
Converting from degrees to radians \((135^{\circ})\left(\Large\frac{\pi}{180^{\circ}}\right) = \Large\frac{135\pi}{180} = \Large\frac{45\cdot3\pi}{45\cdot4}=\Large\frac{3\pi}{4}\)
Example 2:
How many degrees is \(\Large\frac{\pi}{4}\) radians?
Solution 2:
Converting from radians to degrees \(\Large\frac{\pi}{4}\cdot\frac{180^{\circ}}{\pi}\normalsize{=}\Large\frac{180^{\circ}\pi}{4\pi}\normalsize{=}\Large\frac{45^{\circ}\cdot4\pi}{4\pi}\normalsize = 45^{\circ}\)
Another way of solving this conversion is since \(\pi = 180^{\circ}\) we can substitute \(180^{\circ}\) in for \(\pi\). So \(\Large\frac{\pi}{4}\normalsize{=}\Large\frac{180^{\circ}}{4}\normalsize{=}\Large\frac{4\cdot45^{\circ}}{4} \normalsize = 45^{\circ}\)
Arc Length and Radians
We can also find arc length when the measure of the arc is given in radians rather than degrees. We will use our conversion for radians to derive a new arc length formula. Changing the radius, looking at the circumference, and looking at the arc length (in red) we see that the arc length is a fraction of the circumference.
As you recall working with arc length degrees, we can convert to radians:
\(s = \Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\cdot 2\pi r\)
\(\;\;\;=\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\right)\left(\Large\frac{360^{\circ}}{2\pi}\right)\cdot 2\pi r\)
\(\;\;\;=\left(\Large\frac{360^{\circ}}{360^{\circ}}\normalsize\right)\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{2\pi}\right)\cdot 2\pi r\)
So this leads to the equation for arc length:
\(s = \Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\cdot 2\pi r\)
\(\;\;\;=\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\right)\left(\Large\frac{360^{\circ}}{2\pi}\right)\cdot 2\pi r\)
\(\;\;\;=\left(\Large\frac{360^{\circ}}{360^{\circ}}\normalsize\right)\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{2\pi}\right)\cdot 2\pi r\)
So this leads to the equation for arc length:
Arc Length (Arc measure in radians)
\(s = r\cdot \theta\)
\(s\) represents the length of the arc, \(r\) represents the radius of the circle, and \(\theta\) represents the measure of the arc in radians
\(s = r\cdot \theta\)
\(s\) represents the length of the arc, \(r\) represents the radius of the circle, and \(\theta\) represents the measure of the arc in radians
Example 3:
Given \(\bigodot T\) and \(m\angle JTE = \Large\frac{2\pi}{3}\), find the length of \(\overset{\Huge{\frown}}{JE}\) Solution: Since we are given the measure of the central angle, \(\angle JTE\), we know that it is the same as its intercepted arc, \(\overset{\Huge{\frown}}{JE}\). Notice the measure of the central angle (and thus intercepted arc) is given in radians, we can use our new equation to calculate the length of the arc. \(s = r\cdot\theta\) \(s = (4)\left(\Large\frac{2\pi}{3}\right)\) \(s = \Large\frac{8\pi}{3}\) units |
Area of a Sector and Radians
We can also find the area of a sector if the angle measure of the sector is given in radians. As you recall working with arc measure in degrees, we can convert to radians:
Area of sector\( = \Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\cdot\pi r^2\)
\(\;\;\;=\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\right)\left(\Large\frac{360^{\circ}}{2\pi}\right)\cdot\pi r^2\)
\(\;\;\;=\left(\Large\frac{360^{\circ}}{360^{\circ}}\normalsize\right)\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{2\pi}\right)\cdot \pi r^2\)
So this leads to the equation for arc length:
We can also find the area of a sector if the angle measure of the sector is given in radians. As you recall working with arc measure in degrees, we can convert to radians:
Area of sector\( = \Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\cdot\pi r^2\)
\(\;\;\;=\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}\normalsize\right)\left(\Large\frac{360^{\circ}}{2\pi}\right)\cdot\pi r^2\)
\(\;\;\;=\left(\Large\frac{360^{\circ}}{360^{\circ}}\normalsize\right)\left(\Large\frac{m\overset{\Huge{\frown}}{AB}}{2\pi}\right)\cdot \pi r^2\)
So this leads to the equation for arc length:
Area of sector (Arc measure in radians)
\(A = \left(\Large\frac{1}{2}\right)\theta\cdot r^2\)
\(r\) represents the radius of the circle, and \(\theta\) represents the measure of the arc in radians
\(A = \left(\Large\frac{1}{2}\right)\theta\cdot r^2\)
\(r\) represents the radius of the circle, and \(\theta\) represents the measure of the arc in radians
Example 4:
Given \(\bigodot T\) and \(m\angle JTE = \Large\frac{3\pi}{4}\) and \(TJ = 4\) cm, find the area of the sector. Solution: Area of sector \(= \left(\Large\frac{1}{2}\right)\theta\cdot r^2\) \(=\left(\Large\frac{1}{2}\right)\Large\left(\frac{3\pi}{4}\right)\normalsize\cdot4^2\) \(=\Large\frac{3\pi\cdot 16}{8}\) \(=6\pi\) cm\(^2\) |
2) If the arc length of a circle with a radius of \(7.1\) ft is \(26.27\) ft, what is the arc measure that subtends the circle in radians?