Question

1. the circles have a radius of 1, 2, 3, and 4 units, respectively. the diameter divides each circle into two equal sectors. find the arc length of one half of each of these circles. then fill in the table with the indicated values.

1. why does the arc length increase as the radius increases, but the ratio of arc length to the radius remains the same?

Explanation:

Step1: Calculate arc length

The formula for arc length \(s\) is \(s = r\theta\). Since the diameter divides the circle into two equal sectors, the central angle \(\theta=\pi\) radians.

• For \(r = 1\): \(s_1=1\times\pi=\pi\), \(\frac{s_1}{r_1}=\frac{\pi}{1}=\pi\)
• For \(r = 2\): \(s_2=2\times\pi = 2\pi\), \(\frac{s_2}{r_2}=\frac{2\pi}{2}=\pi\)
• For \(r = 3\): \(s_3=3\times\pi=3\pi\), \(\frac{s_3}{r_3}=\frac{3\pi}{3}=\pi\)
• For \(r = 4\): \(s_4=4\times\pi = 4\pi\), \(\frac{s_4}{r_4}=\frac{4\pi}{4}=\pi\)

Step2: Fill in the table

Radius	Arc Length	\(\frac{\text{Arc Length}}{\text{Radius}}\)	Radian Measure of Central Angle
2	\(2\pi\)	\(\pi\)	\(\pi\)
3	\(3\pi\)	\(\pi\)	\(\pi\)
4	\(4\pi\)	\(\pi\)	\(\pi\)

Answer:

Radius	Arc Length	\(\frac{\text{Arc Length}}{\text{Radius}}\)	Radian Measure of Central Angle
2	\(2\pi\)	\(\pi\)	\(\pi\)
3	\(3\pi\)	\(\pi\)	\(\pi\)
4	\(4\pi\)	\(\pi\)	\(\pi\)