Circular Segment Area Calculator

A circular segment is the region enclosed by a chord and the arc connecting the chord's endpoints. Unlike a sector, which extends from the center to the arc along two radii, a segment excludes the triangular region between those radii and the chord. This calculator finds the segment area when you know the circle radius and either the central angle or the chord length. With a radius and central angle, the calculation starts by converting degrees to radians. The sector area is one half times the radius squared times the radian angle. The triangle formed by the two radii has area one half times the radius squared times the sine of that angle. Subtracting the triangle from the sector produces the compact formula: one half times radius squared times the difference between the angle and its sine. When a chord length is known instead of an angle, the calculator first recovers the minor central angle. Half the chord, the radius, and the perpendicular line from the center form a right triangle. The angle is therefore twice the inverse sine of the chord divided by twice the radius. That angle is then used in the same sector-minus-triangle formula. A chord cannot exceed the diameter, so the calculator rejects a chord longer than two radii. The angle method can represent minor or major segments. Angles below 180 degrees produce the smaller region, 180 degrees produces a semicircle, and angles above 180 degrees produce the major segment. The chord method returns the minor segment because a chord by itself does not specify which side of the chord is intended. To find the corresponding major area, subtract the minor result from the full circle area. Circular segments appear in architecture, tank and pipe calculations, window design, machining, landscaping, and geometry exercises. Use one consistent length unit for radius and chord. The resulting area uses the square of that unit, such as square centimeters or square feet. Accurate inputs matter especially for shallow segments, where a small change in chord or angle can noticeably affect area. This calculator uses standard trigonometric relationships and double-precision arithmetic. Displayed values are rounded to make them readable, but the calculation retains more precision internally. Confirm that you need a segment rather than a sector, choose the appropriate input method, and use the formula shown with the result to document your work.