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Subsection 2.6.1 Area of an Annulus

Learner Experience 2.6.3.

\(\textbf{Work in groups }\)
What you require: Circular objects (e.g., two different-sized cups, lids, or rings), ruler or measuring tape and pen and paper (or a calculator).
  1. Look around your surroundings and find two circular objects that can fit inside each other (e.g., two different-sized bowls, two bottle caps, or two CDs).
  2. Place the smaller object inside the larger one to visualize the annulus (ring shape).
  3. Record the values.
    • Measure the radius of the larger circle (\(R\)) from its center to the edge.
    • Measure the radius of the smaller circle (\(r\)) in the same way.
  4. Square both radii. and record your result.
    Subtract the squared radius of the smaller circle from the squared radius of the larger circle.
    Multiply the result by \(\frac{22}{7}\) or \(3.142\)
  5. Discuss with your group how to calculate the area of an annulus.
  6. Try this activity with different circular objects and compare your results.
\(\textbf{Extended Activity}\)

Learner Experience 2.6.4.

\(\textbf{Individual work}\)
Situation: Imagine a running track built around a circular field. The track has an inner boundary (smaller circle) and an outer boundary (larger circle). The track itself forms an annulus.
  • A school is planning to paint the running track.
  • The inner radius of the track is \(30\) meters, and the outer radius is \(35\) meters.
  • The cost of painting is Ksh \(5\) per square meter
  • Calculate:
    1. The area of the track
    2. The total cost of painting the track.

Exploration 2.6.5. Exploring the Area of an Annulus.

An annulus is the region between two concentric circles—one larger circle and one smaller circle that share the same center. In this exploration, you will use sliders to change the outer radius and inner radius of the circles. As the radii change, observe how the shaded ring (the annulus) changes and how its area is calculated.
Before starting, read the instructions below carefully to understand how to use the interactive.

Instructions.
  1. Use the slider labeled R to adjust the outer radius of the larger circle.
  2. Use the slider labeled r to adjust the inner radius of the smaller circle.
  3. Observe how the shaded region between the circles changes as the radii change.
  4. Look at the calculation panel below the diagram.
    • The area of the outer circle,
    • The area of the inner circle, and
    • The area of the annulus.
  5. Notice how the area of the annulus is computed by subtracting the inner circle’s area from the outer circle’s area.
Use the interactive to investigate the following questions:
  1. 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?
  2. Slowly increase the outer radius while keeping the inner radius fixed. What happens to the annulus area? Why do you think this occurs?
  3. Keep the outer radius fixed and increase the inner radius. What happens to the annulus area as the inner circle grows larger?
  4. 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?
  5. 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.
  6. 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.
Figure 2.6.6. Interactive: Area of an Annulus

Key Takeaway 2.6.7.

An \(\textbf{annulus}\) is the region between two concentric circles that share the same center but have different radii as shown below.
The area of an annulus (a ring-shaped object) is found by subtracting the area of the smaller, inner circle from the area of the larger, outer circle.
The formula is:
\begin{align*} A_\text{annulus} = \amp A_\text{outer circle} - A_\text{inner circle} \\ = \amp \pi R^2 - \pi r^2 \\ = \amp \pi (R^2 - r^2) \end{align*}
Where:
  • \(R\) is the radius of the outer circle
  • \(r\) is the radius of the inner circle

Example 2.6.8.

Find the area of an the annulus drawn below;
Solution.
\(R=10\,cm\) and \(r=6\,cm\)
\begin{align*} A= \amp \pi(R^2-r^2) \\ =\amp \pi(10^2-6^2) \\ =\amp \pi(100-36)\\ =\amp 64\pi \\ =\amp 64 \times \frac{22}{7}\\ = \amp 201.06 \,cm^2 \end{align*}

Example 2.6.9.

A wheel has an outer radius of \(40 \,cm\text{,}\) and its inner hub has a radius of \(10 \,cm\text{.}\) Find the area of the wheel’s annular region.
Solution.
The outer radius of the wheel \(= 40\,cm\text{.}\)
Inner hub radius \(=10\,cm\)
\begin{align*} A= \amp \pi(R^2-r^2)\\ =\amp \frac{22}{7}(40^2-10^2) \\ =\amp \frac{22}{7}(1\,600-100)\\ =\amp \frac{22}{7} \times 1\,500 \\ =\amp 4\,712.39 \,cm^2 \end{align*}
\(\textbf{Exercises}\)

Exercises Exercises

1.

A ring-shaped garden has an outer radius of \(12\) meters and an inner radius of \(7\) meters. Find the area of the garden.

2.

A circular tabletop has a hole in the middle for an umbrella. The outer radius of the table is \(1.5 \,m\text{,}\) and the hole has a radius of \(0.5 m\) as shown below.
Find the area of the tabletop.

3.

A circular swimming pool has an outer radius of \(8\) meters, and a smaller circular island is in the center with a radius of \(2\) meters. Find the area of the water surface.

4.

A steel pipe has an outer diameter of \(80\) units and an inner diameter of \(60\) units, what is the area of the cross-section?

5.

Find the area of the figure below; use \(\pi=3.142\)

6.

What is the area of the annulus shown; (Leave your answer to \(3\) significant figures).

Checkpoint 2.6.10.

Checkpoint 2.6.11.

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