An annulus is the flat ring-shaped region between two circles that share the same center. It appears in washers, circular tracks, pipe cross sections, gears, and many design patterns. Finding its area matters because many real objects are made by removing a smaller circle from a larger one.
The main idea is simple: subtract the inner circle area from the outer circle area.
If the outer radius is R and the inner radius is r, the annulus area is A = πR^2 - πr^2, which can be written as A = π(R^2 - r^2). The radii must be measured from the common center, not across the ring. When the ring has a constant thickness t, the relationship is R = r + t.
This lets you solve many problems from a diagram, a word problem, or measurements of diameters.
Key Facts
- Annulus area: A = π(R^2 - r^2), where R is the outer radius and r is the inner radius.
- Circle area formula: A = πr^2.
- Outer circle area: A_outer = πR^2.
- Inner circle area: A_inner = πr^2.
- Ring thickness: t = R - r.
- If diameters are given, use R = D_outer/2 and r = D_inner/2 before calculating area.
Vocabulary
- Annulus
- An annulus is the ring-shaped region between two concentric circles.
- Concentric circles
- Concentric circles are circles that share the same center point.
- Outer radius
- The outer radius is the distance from the common center to the outside circle.
- Inner radius
- The inner radius is the distance from the common center to the inside circle.
- Ring thickness
- Ring thickness is the difference between the outer radius and the inner radius.
Common Mistakes to Avoid
- Subtracting radii before squaring, A = π(R - r)^2, is wrong because area depends on the square of each radius, so the correct formula is A = π(R^2 - r^2).
- Using diameters as radii is wrong because the radius is half the diameter, so always divide each diameter by 2 first.
- Measuring the inner radius from the edge of the outer circle is wrong because both radii must start at the common center.
- Forgetting square units is wrong because area is measured in units such as cm^2, m^2, or in^2, not just cm, m, or in.
Practice Questions
- 1 An annulus has outer radius 10 cm and inner radius 6 cm. Find its area in terms of π and as a decimal using π ≈ 3.14.
- 2 A metal washer has an outer diameter of 18 mm and an inner diameter of 8 mm. Find the area of the washer in mm^2 using π ≈ 3.14.
- 3 Two rings have the same thickness of 2 cm. Ring A has inner radius 3 cm, and Ring B has inner radius 8 cm. Explain which ring has the larger area and why.