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 (between \(0^\circ\) and \(180^\circ\)). -
Observe the shaded region. This is the circular segment.
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Notice the triangle drawn inside the sector. The segment is formed by subtracting this triangle from the sector.
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Look at the calculation panel, which shows:
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The area of the sector,
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The area of the triangle, and
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The resulting area of the segment.
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Use the interactive to investigate the following questions:
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Set the angle to about \(60^\circ\text{.}\) How does the shaded segment compare to the sector? Which part is being removed to form the segment?
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Increase the angle gradually from \(30^\circ\) to \(150^\circ\text{.}\) How does the shape of the segment change? What happens to its area?
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At what angle does the segment appear largest? Explain your reasoning based on what you observe.
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Keep the angle fixed and increase the radius. How does the segment area change? What does this suggest about the role of the radius in the formula?
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Compare the sector area and triangle area for different angles. How does their difference determine the segment area?
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When the angle is very small, what happens to the segment? What does this suggest about the relationship between the arc and the chord?
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Based on your observations, how would you describe a formula for the area of a segment using the sector area and the triangle area?