Calculus How To

Annulus: Definition, Area

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The annulus is the region between two concentric rings—two circles that share the same center. This flat donut shape, or washer, is important for many engineering applications.

The center of mass for an annulus is not within the shape, but at the shared center of both circles. In the image below, the center of mass is labeled as “C”.
center of mass annulus

Area of an Annulus

To calculate the area, think about the shape as a larger circle with a smaller circle taken out of it. If the radius of the larger circle is R and the radius of the smaller circle is r, the radius of the circle will be π(R2 – r2). In the following image, the large ring has a radius of 4 and the smaller ring has a radius of 2:

You can get to this same result using Calculus if you imagine dividing up your annulus into many many annuli (very thin donuts) with an infinitesimal width. The formula is:


Here, you integrate over very thin rings of width dρ, where ρ is the radius of the inner ring. Notice the result is the same as that which you’d reach by using the simple geometrical method above.

Since the full circle is 360 degrees, the area of a M degree piece of an annulus would be given by:
π(R2 – r2)(M/360).

Let’s find the area of a 90 degree slice—a quarter—of an annulus where the radius of the outer ring is 2 and the radius of the inner circle is 1. Plugging those numbers into the formula above, we find our area is:
π(22 – 12)(90/360), or  π(3)(90/360).
This comes out to 3π/4.

An Annulus in Complex Analysis

In complex analysis, you can describe an annulus by writing ann(a; r, R), where:

  • a is the center,
  • r is the inner radius,
  • R is the outer radius.

It is defined as an open region such that

r < |z-a| < R

An annulus with an inner radius of zero is called a punctured disk; it is a filled circle with the (infinitesimally small) central point missing.


Page, John. Calculating the Area of…. Retrieved from on September 28, 2018.
Weisstein, Eric W. “Annulus.” From MathWorld–A Wolfram Web Resource. Retrieved from on September 28, 2018

Stephanie Glen. "Annulus: Definition, Area" From Calculus for the rest of us!

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