Geometry

# Circles - Central Angles

In the above diagram, if the radius of the circle is $$18$$ and the angle $$AOB$$ (central angle) is $${30}^\circ,$$ what is the measure of the arc $$\widehat{AB} ?$$

In the above diagram, if angle $$COD$$ is $$\theta_2={123}^\circ$$ and the measures of arc $$\widehat{AB}$$ and arc $$\widehat{CD}$$ are $$5\text{ cm}$$ and $$15\text{ cm},$$ respectively, what is the angle $$\theta_1 (= \angle AOB)?$$

In the above diagram, $$\overline{AB}$$ is a diameter of the circle centered at $$O$$, $$\overline{AB} \parallel \overline{DC}$$, and $$\angle BOC=\theta={30}^\circ.$$ If the measure of the arc $$\widehat{BC}$$ is $$l_1=12\pi\text{ cm},$$ what is the measure of the arc $$\widehat{CD} ?$$

In the above diagram, $$\overline{AB}$$ is a diameter of the circle centered at $$O,$$ and $$\lvert{\overline{OD}}\rvert=\lvert{\overline{DE}}\rvert$$ and $$\angle E=\theta={15}^\circ.$$ If the measure of the arc $$\widehat{BD}$$ is $$5\text{ cm},$$ what is the measure of the arc $$\widehat{AC}$$ in $$\text{cm} ?$$

Imagine a circle divided into equal sectors via $$n$$ straight lines that cross point $$O$$, the center of the circle. Let sector $$ABO$$ be one of those sectors. When $$n=20,$$ what is $$\angle OAB$$ in degrees? (Round down to the nearest degree if there is a decimal.)

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