Instructions.
-
Use the slider labeled
Outer Radius (R)to change the size of the larger circle. -
Use the slider labeled
Inner Radius (r)to change the size of the inner circle. -
Use the slider labeled
Central Angle (ΞΈ)to adjust the size of the angle that forms the sector. -
Observe the shaded region between the two arcs. This region is the annular sector.
-
Look at the calculation panel below the diagram. It shows:
-
The area of the full annulus,
-
The fraction of the circle determined by the angle \(\theta/360\text{,}\) and
-
The resulting area of the annular sector.
-
Use the interactive to investigate the following questions:
-
Set the central angle to about \(90^\circ\text{.}\) What fraction of the full annulus does the shaded region represent?
-
Keep the radii fixed and increase the central angle from \(30^\circ\) to \(180^\circ\text{.}\) How does the shaded region change? What happens to the calculated area?
-
Fix the central angle and increase the outer radius while keeping the inner radius the same. How does this affect the size of the annular sector?
-
Keep the outer radius fixed and increase the inner radius. What happens to the thickness of the ring and the area of the annular sector?
-
Try making the inner radius very small. What does the shaded region begin to resemble? How does this relate to the area of a regular sector of a circle?
-
Based on your observations, how could you write a formula for the area of an annular sector using the outer radius \(R\text{,}\) the inner radius \(r\text{,}\) and the angle \(\theta\text{?}\)
-
Compare the area of the annular sector with the area of the entire annulus. How does the ratio \(\theta/360\) determine the portion of the annulus that is shaded?