Annulus geometry: rings, washers and pipe walls
How the area of a ring is built from two circles, why the difference of squares matters, and where annulus math is used.
A circle with a circle removed
The cleanest way to picture an annulus is as a large disc with a smaller disc cut out of its exact centre. Because the area of any disc is pi times its radius squared, the ring that remains has an area of pi times the outer radius squared minus pi times the inner radius squared. Factoring out pi leaves the compact form pi times the quantity (R squared minus r squared). That single subtraction is all the area formula ever does, no matter how thick or thin the ring is.
Why the difference of squares matters
The expression R squared minus r squared is a difference of squares, which factors into (R plus r) times (R minus r). The second factor, R minus r, is exactly the width of the ring that this tool also reports. This gives a useful intuition: a ring's area grows with both its average radius and its width, so a wide ring near the rim of a large circle encloses far more area than a thin ring near the centre. It also explains why doubling the width of a narrow ring roughly doubles its area, while the radius is large compared to the width.
Two edges, two circumferences
Unlike a solid disc, an annulus has an inside edge and an outside edge, so it carries two circumferences. The outer circumference is 2 times pi times the outer radius and the inner circumference is 2 times pi times the inner radius. These lengths matter in practice: the outer one might be the amount of trim around a washer, while the inner one is the opening a bolt passes through. Reporting both means the ring is fully described by its area, its width and the length of each of its two boundaries.
Where ring area shows up
Annulus calculations appear whenever something is round with a hole in the middle. The cross-section of a pipe wall is an annulus, so its area drives how much metal a length of pipe contains and how much fluid the bore carries. Flat washers, gaskets, O-rings, CD and vinyl records, running tracks, and the sensor area of a ring-shaped detector are all annuli. Even in calculus the method of finding volumes by washers slices a solid into thin annular discs and sums their areas, which is the same formula used here applied over and over.