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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?

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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