Area of Crescent Calculator

This calculator finds the area between two concentric circles, a shape more precisely called an annulus or circular ring. The legacy page describes it as a crescent. When both circles share a center, calculate the region by subtracting the smaller circle from the larger one: A = pi × (R² - r²). Here R is the outer radius and r is the inner radius. The outer radius must be larger because the inner circle is removed from inside it. For an outer radius of 5 and an inner radius of 3, the difference of the squared radii is 25 - 9 = 16. Multiplying 16 by pi gives about 50.2655 square units. Subtracting the radii before squaring would be incorrect: pi × (R - r)² gives the area of a separate circle whose radius equals the ring width, not the area of the ring. The difference-of-squares identity can also express the correct formula as pi × (R - r) × (R + r). Ring width is the radial thickness R - r. The calculator also reports total boundary length, which is the outer circumference plus the inner circumference: 2 × pi × R + 2 × pi × r. This boundary is useful when estimating trim around both edges, while area is useful for material covering the region. A washer, gasket, circular track, pipe cross-section, and ring-shaped garden bed can all be modeled this way. Both radii need the same linear unit. If they are measured in millimeters, width and boundary are in millimeters while area is in square millimeters. Convert unmatched units before entering them. The calculator leaves the unit label to you so it can work with centimeters, meters, inches, feet, or any other consistent unit. Zero is allowed for the inner radius; in that special case, there is no hole and the result is simply the area of a full circle. A true offset crescent formed by overlapping circles with different centers requires the distance between their centers and a circular-segment formula. This calculator intentionally follows the source page's concentric ring definition, so it does not model an offset lunar crescent. Confirm that your diagram shows a common center before applying this result. For nonconcentric circles, use an overlap or lune calculation instead. The calculation uses a precise built-in value of pi and rounds only the displayed answer. Keep the full value for subsequent calculations when possible. Physical measurements have tolerances, and ring area can change noticeably with radius because both values are squared. Measure the outer and inner radii carefully, use the same reference center, and round the final area according to the precision of those measurements.