Math Solver
Annulus
Studio
06.12 • Circle Geometry

Annulus

Work with the ring-shaped region between two concentric circles and connect that shape to thickness, area difference, and shared center.

Interactive diagram Live labels and measurements Worked examples PNG graph downloads
Annulus
Interactive diagram

Annulus Diagram

Resize the inner and outer circles while keeping the same center so the ring region stays a true annulus.

Use the movable diagram to see what defines annulus, how the labels relate to the figure, and what stays true as the board changes.

Definition: An annulus is the ring-shaped region between two concentric circles.
Detailed definition

Understanding Annulus

An annulus is the region between two coplanar concentric circles. It looks like a flat ring because the inner circle removes a smaller disk from a larger one without shifting the center.

The geometry of an annulus comes from subtraction. Its area is found by subtracting the area of the inner disk from the area of the outer disk, which is why the two circles must share a center.

This page keeps both circles aligned around the same center so you can see that an annulus is defined by concentric structure, not merely by having one circle inside another.

Key facts

Important ideas to remember

  • An annulus is the ring-shaped region between two concentric circles.
  • Both circles in an annulus are concentric, meaning they share the same center.
  • The annulus has an inner radius and an outer radius, and both matter in measurement problems.
  • Annulus area is the difference between two circle areas, not the product or the average of the two radii.
Where it is used

Where annulus shows up

  • Use annuli when working with washers, rings, circular tracks, and pipe cross-sections.
  • Use them in area problems where one circular region is removed from a larger one.
  • Use the concept whenever a circle-based figure has a hole in the middle but keeps the same center.
Common mistakes

What to watch out for

  • Do not call the region an annulus if the two circles are not concentric.
  • Do not subtract radii and mistake that for the area of the ring.
  • Do not confuse the thickness of the annulus with its full area.
Worked examples

Annulus examples

Use these worked examples to see the idea in a clean diagram first, then in the kind of reasoning students usually need for classwork, homework, or test practice.

Example 1

Example 1: Tracing the boundary of annulus

Identify exactly which arcs, radii, or chords form the region before talking about measurement.

  • Mark the endpoints first.
  • Trace the boundary pieces in order.
  • Name the region only after the border is clear.

Result: The region is easier to classify because its full boundary has been read carefully.

Example 2

Example 2: Using annulus in a measurement setting

Turn the highlighted region into the correct area or arc-length question rather than choosing a formula too early.

  • Read which part of the circle is highlighted.
  • Select the matching measurement idea.
  • Check that the boundary of the region matches the formula you chose.

Result: The measurement stays connected to the actual part of the circle being studied.

For

Why this page helps

This page helps because annulus depends on concentric circles, not just on two circles drawn near each other. The shared center is what makes the ring region valid and what supports its area formula.

Do

What you can do here

  • Resize the inner and outer circles and keep track of how the ring region changes.
  • Compare outer radius, inner radius, and annulus thickness on the same figure.
  • Save a diagram that clearly shows why shared center is essential for an annulus.
Learning outcome

What this page helps you do

These takeaways are meant to help you recognize the idea faster, read diagrams more accurately, and use the topic with more confidence in real problems.

1

Annulus

Recognise annulus as a concentric-circle region rather than as a loose ring shape.

2

Annulus

Set up annulus area problems with stronger structural understanding.

3

Annulus

Use inner-radius and outer-radius language more precisely in circle measurement work.

06

Back to Circle Geometry

Return to the category page to open another concept in circle geometry.

ST

Geometry Construction Studio

Use a dedicated geometry drawing board for points, segments, rays, lines, angles, circles, triangles, rectangles, pencil sketches, and virtual measuring tools.

06.11

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A circle segment is the region between a chord and its arc.

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A central angle has its vertex at the center of the circle.