Ring (annulus)

Properties

• Its width is the difference of the two radii.
• Its area is the difference of the areas of the two circles.
• It has two edges: the outer circle and the inner circle.

Why the formulas work

Take the big circle and remove the small one from its middle. What is left has area πR² − πr², which is tidied to π(R² − r²). A useful form is π(R + r)(R − r): the width of the ring times the length of the circle running along its middle.

Worked example

Take a ring (annulus) with outer radius 14 cm and inner radius 7 cm.

1. Area. Area = π(R² − r²). Here: π × (14² − 7²) = 461.81 cm² (with π = 22/7: 462 cm²).
2. Total edge. Outer edge + inner edge = 2π(R + r). Here: 2 × π × (14 + 7) = 131.95 cm (with π = 22/7: 132 cm).
3. Width. Width = R − r. Here: 14 − 7 = 7 cm.

Finding a missing measurement

If the area and the inner radius are known, R² = r² + A ÷ π. An inner radius of 7 cm and an area of 462 cm² give R² = 49 + 147 = 196, so R = 14 cm and the ring is 7 cm wide.

Mistakes to avoid

• Subtracting the radii first and then squaring: π(R − r)² is wrong.
• For a path of width w outside a circle of radius r, the outer radius is r + w, not w.

Where you see it

A circular path round a fountain, a washer, a bangle laid flat, a CD and the cross-section of a pipe are rings.

Try these

Work each one out, then tap to check.

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