Grade 10 Mathematics Teachers’ Book

Use the slider labeled R to adjust the outer radius of the larger circle.

Use the slider labeled r to adjust the inner radius of the smaller circle.

Observe how the shaded region between the circles changes as the radii change.

Look at the calculation panel below the diagram.

Notice how the area of the annulus is computed by subtracting the inner circle’s area from the outer circle’s area.

Start with an outer radius of about \(8\) and an inner radius of about \(4\text{.}\) How does the shaded annulus compare visually to the areas of the two circles?

Slowly increase the outer radius while keeping the inner radius fixed. What happens to the annulus area? Why do you think this occurs?

Keep the outer radius fixed and increase the inner radius. What happens to the annulus area as the inner circle grows larger?

Try making the inner radius very small. What does the annulus begin to look like? What does this suggest about the relationship between the annulus area and the area of a circle?

Can you write a formula for the area of an annulus using the radii \(R\) and \(r\text{?}\) Compare your formula with the calculations shown in the panel.

If two annuli have the same difference between their radii (for example, \(R - r = 2\)), do they always have the same area? Use the sliders to test your conjecture.