Question

1. two concentric circles (circles with the same center) have radii of 5 cm and 8 cm. find the area of the ring (annulus) between them. leave answer in terms of \\(\pi\\)

part 8: reflection questions

1. explain in your own words why the arc length formula includes the fraction \\(\theta/360^\circ\\).

Explanation:

Step1: Recall the formula for the area of an annulus

The area of an annulus (the region between two concentric circles) is given by the formula \( A=\pi R^{2}-\pi r^{2} \), where \( R \) is the radius of the larger circle and \( r \) is the radius of the smaller circle.

Step2: Identify the radii of the two circles

From the problem, the radii of the two concentric circles are \( R = 8\space\text{cm} \) and \( r=5\space\text{cm} \).

Step3: Substitute the values of \( R \) and \( r \) into the formula

Substitute \( R = 8 \) and \( r = 5 \) into the formula \( A=\pi R^{2}-\pi r^{2} \). We get \( A=\pi(8)^{2}-\pi(5)^{2} \).

Step4: Simplify the expression

First, calculate \( 8^{2}=64 \) and \( 5^{2} = 25 \). Then the expression becomes \( A=\pi\times64-\pi\times25 \). Factor out \( \pi \), we have \( A=\pi(64 - 25) \). Calculate \( 64-25=39 \). So \( A = 39\pi \space\text{cm}^2 \).

Answer:

The area of the ring (annulus) between the two concentric circles is \( \boldsymbol{39\pi}\space\text{square centimeters} \).