Annulus Calculator
How It Works
An annulus (plural: annuli) is the region between two concentric circles — essentially a ring shape. The area formula is A = pi(R^2 - r^2), where R is the outer radius and r is the inner radius. This can also be written as A = pi(R+r)(R-r).
Annuli appear in many real-world applications: pipe cross-sections, washers, O-rings, circular tracks, and the area between a plate and a bowl on a table. The factored form A = pi(R+r)(R-r) shows that the area depends on both the average radius and the width of the ring.
An interesting property: the area of an annulus equals the area of a circle whose diameter is the chord of the outer circle that is tangent to the inner circle. This surprising result is independent of the inner and outer radii individually.
Frequently Asked Questions
What is the formula for annulus area?
Area = π(R² - r²), where R is the outer radius and r is the inner radius. This can be factored as π(R+r)(R-r). For example, with R=10 and r=6: Area = π(100-36) = 64π ≈ 201.06.
How do I calculate the area of a pipe cross-section?
A pipe cross-section is an annulus. Use the outer diameter/2 for R and inner diameter/2 for r. For a 2-inch pipe with 1/8-inch walls: R=1, r=0.875, Area = π(1²-0.875²) ≈ 0.736 sq inches.
What is the difference between a ring and an annulus?
In mathematics, "annulus" is the formal term for the region between two concentric circles. "Ring" is the everyday term. In engineering, "ring" may also refer to the physical object (like an O-ring), while "annulus" refers to the geometric shape.
Can the inner radius be zero?
If the inner radius is zero, the annulus becomes a full circle. The formula still works: Area = π(R²-0²) = πR². This is just the standard circle area formula.