Central Angle Calculator

A central angle is an angle whose vertex sits at the exact center of a circle. Its two rays extend from the center to points on the circumference, enclosing an arc between them. This calculator finds that angle when the arc length and circle radius are known. It reports both radians, the natural unit for circle calculations, and degrees, the familiar unit used in geometry diagrams and practical measurements. The calculation follows the defining relationship s = rθ, where s is arc length, r is radius, and θ is the central angle in radians. Rearranging gives θ = s ÷ r. Because the arc length and radius are lengths, they must use the same unit. You may enter centimeters, meters, inches, or any other consistent unit; those units cancel during division, leaving a dimensionless radian measure. The calculator then multiplies radians by 180 ÷ π to obtain degrees. A full turn around a circle contains 2π radians, or 360 degrees. A semicircle therefore has a central angle of π radians, or 180 degrees, and a quarter circle has π ÷ 2 radians, or 90 degrees. These familiar cases are useful for checking results. An arc equal in length to the radius always subtends exactly one radian, approximately 57.2958 degrees. Central angles appear in surveying, mechanical design, architecture, astronomy, mapping, and classroom geometry. They help determine sector areas, chord lengths, wheel rotation, curved travel distances, and the placement of repeated features around circular objects. Once the central angle is known, sector area can be found with one half times radius squared times the angle in radians. For the most reliable answer, measure the arc along the circumference rather than using the straight-line chord between its endpoints. Also confirm that both entered measurements describe the same circle and share one unit. The calculator accepts decimal values and keeps enough precision for typical educational, engineering, and construction work while presenting a readable rounded result.