Annulus Area

Area of a circular ring (annulus)

A circular ring is the band left over between two concentric circles, one with outer radius R and one with inner radius r, where R > r. To get its area you subtract the smaller disc from the larger one: A = π(R² - r²). That difference of squares factors into A = π(R + r)(R - r), which comes in useful when the thickness (R - r) is thin and is really the number you care about. With R = 10 and r = 4 you get A = π(100 - 16) = 84π ≈ 263.89.

Applications

You run into the ring shape all over the place: flat washers, the painted circles marked on roads, the cross-section of a hollow pipe (handy for working out mass per metre), even a ring-cut slice of pizza. They're all the same thing underneath, a flat shape with a round hole and a width that stays the same the whole way around.

FAQ

What if R = r? Then there's no ring left at all, just a circle with no thickness, so the area works out to zero. The formula only makes sense when R > r.

Can I use diameters instead? Sure. If you measured the outer diameter D and inner diameter d, the area is A = π(D² - d²)/4.

What if the circles are not concentric? Then you don't really have a ring anymore. The width varies as you go around, so it's an off-centre shape. The area still equals the difference of the two disc areas, but only as long as the smaller circle sits entirely inside the larger one.