Question

consider circle c with radius 5 cm and a central angle measure of 60°.
what fraction of the whole circle is arc rs?
is the approximate circumference of the circle?
is the approximate length of arc rs?

Explanation:

First Question: What fraction of the whole circle is arc RS?

Step1: Recall the total degrees in a circle

A full circle has \( 360^\circ \).

Step2: Calculate the fraction

The central angle for arc RS is \( 60^\circ \). The fraction is \( \frac{60^\circ}{360^\circ} \).
Simplify \( \frac{60}{360}=\frac{1}{6} \).

Step1: Use the circumference formula

The formula for the circumference of a circle is \( C = 2\pi r \), where \( r = 5 \) cm.

Step2: Substitute the value of \( r \)

\( C = 2\times\pi\times5 = 10\pi \approx 10\times3.14 = 31.4 \) cm.

Step1: Recall the arc length formula

The length of an arc is given by \( \text{Arc Length}=\text{Fraction of the circle}\times\text{Circumference} \).
We know the fraction is \( \frac{1}{6} \) and the circumference is approximately \( 31.4 \) cm.

Step2: Calculate the arc length

\( \text{Arc Length}=\frac{1}{6}\times31.4\approx\frac{31.4}{6}\approx5.23 \) cm.
Or using the formula \( \text{Arc Length}=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta = 60^\circ \), \( r = 5 \) cm.
\( \text{Arc Length}=\frac{60^\circ}{360^\circ}\times2\times\pi\times5=\frac{1}{6}\times10\pi\approx\frac{31.4}{6}\approx5.23 \) cm.

Answer:

\( \frac{1}{6} \)

Second Question: What is the approximate circumference of the circle?