Question

consider circle o, in which arc xy measures 16π cm. the length of a radius of the circle is 32 cm. what is the circumference of the circle? what is the ratio of the arc length to the circumference? what is the measure of central - angle xoy?

Explanation:

Step1: Recall circumference formula

The formula for the circumference of a circle is $C = 2\pi r$. Given $r = 32$ cm, then $C=2\pi\times32 = 64\pi$ cm.

Step2: Calculate arc - to - circumference ratio

The arc length $l = 16\pi$ cm and the circumference $C = 64\pi$ cm. The ratio $\frac{l}{C}=\frac{16\pi}{64\pi}=\frac{1}{4}$.

Step3: Find central - angle measure

The ratio of the central angle $\theta$ (in radians) to $2\pi$ (the total angle in a circle) is equal to the ratio of the arc length to the circumference. Let the central angle be $\theta$. We know $\frac{\theta}{2\pi}=\frac{1}{4}$, so $\theta=\frac{2\pi}{4}=\frac{\pi}{2}$ radians. To convert to degrees, use the conversion factor $\frac{180^{\circ}}{\pi}$. Then $\theta=\frac{\pi}{2}\times\frac{180^{\circ}}{\pi}=90^{\circ}$.

Answer:

1. $64\pi$
2. $\frac{1}{4}$
3. $90^{\circ}$