Question

1. each circle has a different radius. fill in the chart with the indicated values.
2. what is the radian measure of an angle that measures 45°?

Explanation:

Step1: Calculate arc length formula

The formula for arc length \(s\) is \(s=\frac{\theta}{360}\times2\pi r\) (where \(\theta = 45^{\circ}\)).
For \(r = 10\):

$$
LATEXBLOCK0
$$

For \(r = 20\):

$$
LATEXBLOCK1
$$

For \(r = 30\):

$$
LATEXBLOCK2
$$

Step2: Calculate \(\frac{\text{Arc Length}}{\text{Radius}}\)

For \(r = 10\), \(\frac{2.5\pi}{10}=0.25\pi\) (but if we use \(s = r\theta\) in radians, \(\theta=\frac{\pi}{4}\approx 0.785\), using the non - \(\pi\) form of arc length \(s = 2.5\pi\approx7.85\), \(\frac{s}{r}=\frac{7.85}{10} = 0.785=\frac{\pi}{4}\)). Using the formula \(\frac{s}{r}=\theta\) (in radians).
For \(r = 20\), \(s = 5\pi\approx15.7\), \(\frac{s}{r}=\frac{15.7}{20}=0.785=\frac{\pi}{4}\)
For \(r = 30\), \(s=7.5\pi\approx23.55\), \(\frac{s}{r}=\frac{23.55}{30}=0.785=\frac{\pi}{4}\)

Step3: Calculate area of sector formula

The formula for the area of a sector \(A=\frac{\theta}{360}\times\pi r^{2}\)
For \(r = 10\):

$$
LATEXBLOCK3
$$

For \(r = 20\):

$$
LATEXBLOCK4
$$

For \(r = 30\):

$$
LATEXBLOCK5
$$

Step4: Convert \(45^{\circ}\) to radians

We know that to convert degrees to radians, we use the formula \(\text{Radians}=\frac{\pi}{180}\times\text{Degrees}\)
For \(\theta = 45^{\circ}\), \(\text{Radians}=\frac{\pi}{180}\times45=\frac{\pi}{4}\)

Answer:

Radius	Arc Length	\(\frac{\text{Arc Length}}{\text{Radius}}\)	Area of Sector
20	\(5\pi\)	\(\frac{\pi}{4}\)	\(50\pi\)
30	\(7.5\pi\)	\(\frac{\pi}{4}\)	\(112.5\pi\)

The radian measure of a \(45^{\circ}\) angle is \(\frac{\pi}{4}\) radians.