Area of Annulus
Compute the area of a circular ring (annulus) from outer and inner radii. Free geometry calculator with circumferences, average radius, and ring width.
About This Calculator
This area of annulus calculator computes the area of a circular ring (annulus) -- the region between two concentric circles -- given the outer radius (R) and inner radius (r). It also provides the outer circumference, inner circumference, average radius, and ring width.
The formula used is A = pi(R^2 - r^2), which subtracts the area of the inner circle from the area of the outer circle. This formula applies to any circular ring regardless of the units used -- simply enter both radii in the same unit system.
How to Use
Enter the outer radius and inner radius in any consistent unit (cm, inches, feet, meters). Click "Calculate Annulus Area" to get the ring area, both circumferences, average radius, and ring width instantly. All inputs are saved to the URL for easy sharing.
Applications
Annulus area calculations are used in mechanical engineering (pipe cross-sections, washers, gaskets), construction (circular columns, ring beams), manufacturing (ring-shaped parts), geometry education, and everyday tasks like cutting ring-shaped templates or measuring donut-shaped objects.
Frequently Asked Questions
What is the formula for the area of an annulus?
The area of an annulus (circular ring) is calculated using the formula A = pi(R^2 - r^2), where R is the outer radius and r is the inner radius. This gives the area of the ring-shaped region between the two concentric circles.
What is the difference between an annulus and a circle?
A circle is a closed curve with all points at the same distance from the center. An annulus (also called a circular ring) is the region between two concentric circles -- like a washer, a ring, or a donut shape. It has an outer radius (R) and a smaller inner radius (r).
How do you find the circumference of an annulus?
An annulus has two circumferences: the outer circumference C_outer = 2piR and the inner circumference C_inner = 2pir. There is no single circumference -- the annulus is bounded by two circles.
What real-world objects are shaped like an annulus?
Common examples include washers, rings, donuts, CDs and DVDs, the cross-section of a pipe wall, crop circles, frisbees with a center hole, and the circular tracks on a running field. In architecture, annulus shapes appear in domes, archways, and circular windows.
Can the inner radius be equal to the outer radius?
No. The inner radius must be strictly less than the outer radius. If r = R, the annulus area is zero -- there is no ring. If r > R, it is not a valid annulus. The calculator will show an error message if the inner radius is not smaller than the outer radius.
How is the area of an annulus used in engineering?
In engineering, the annulus area is used to calculate the cross-sectional area of pipes (the wall area between inner and outer diameter), the material needed for manufacturing rings and gaskets, the flow area in annular ducts, and the moment of inertia for circular shafts with a bore.
What is the average radius of an annulus?
The average radius of an annulus is (R + r) / 2, the arithmetic mean of the outer and inner radii. It represents the midpoint of the ring width. The ring width itself is R - r.
Is this annulus area calculator free to use?
Yes, this annulus area calculator is completely free to use. No registration or download is required, and you can calculate as many annuli as you like with shareable URL links.