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Properties of Circles

What is pi? If the area of a circle is 49π, what is its circumference? If a circle is inscribed in a square, what is the ratio of the areas of the circle and the square?

Central Angle

         

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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