Multiple Choice
If the measure of central angle in a circle of radius units is radians, what is the area of the shaded sector?
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1. Measuring Angles
Radians
Multiple Choice
If the measure of central angle RST is radians in a circle of radius , what is the area of the shaded sector formed by this angle?
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Verified step by step guidance1
Recall that the area of a sector of a circle is a fraction of the total area of the circle, where the fraction is determined by the ratio of the central angle to the full angle of the circle (which is \(2\pi\) radians).
The total area of a circle with radius \(r\) is given by the formula \(\pi r^{2}\).
Since the central angle \(\theta\) is in radians, the fraction of the circle's area that the sector occupies is \(\frac{\theta}{2\pi}\).
Multiply the total area of the circle by this fraction to find the area of the sector: \(\text{Area of sector} = \pi r^{2} \times \frac{\theta}{2\pi}\).
Simplify the expression by canceling \(\pi\) in numerator and denominator, resulting in the formula for the area of the sector: \(\frac{\theta r^{2}}{2}\).
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