Instructions.
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Use the
Radius (r)slider to change the size of the circle. -
Use the
Angle (ΞΈ)slider to adjust the central angle of the sector. -
Observe how the shaded sector changes as the angle increases or decreases.
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Look at the calculation panel below the diagram. It shows:
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The area of the full circle,
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The fraction of the circle represented by the angle \(\theta/360\text{,}\) and
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The resulting sector area.
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Notice how the sector area changes when either the radius or the angle changes.
Use the interactive to explore the following questions:
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Set the angle to about \(90^\circ\text{.}\) What fraction of the circle does this sector represent? Compare the sector area with the full circle area shown in the panel.
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Keep the radius fixed and increase the angle gradually from \(30^\circ\) to \(180^\circ\text{.}\) How does the sector area change? What pattern do you notice?
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Set the angle to \(360^\circ\text{.}\) What does the sector become? What does the sector area equal in this case?
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Fix the angle at \(60^\circ\text{.}\) Now increase the radius. How does changing the radius affect the sector area?
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Compare two sectors with the same radius but different angles. How does the ratio \(\theta/360\) relate to the size of the sector?
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Based on your observations, write a formula for the area of a sector in terms of the radius \(r\) and the angle \(\theta\text{.}\) How does your formula match the calculation shown in the panel?