Segment Area Calculator

A circular segment is the region bounded by a chord and the arc it cuts off on a circle. If you imagine slicing a circle with a straight line, the smaller of the two resulting pieces is the circular segment. When the chord passes through the center, each piece is a semicircle.

When you know the central angle θ (in radians) and radius r, the segment area is A = r² × (θ − sin θ) / 2. When you know the segment height h and radius r instead, use A = r² × arccos((r − h) / r) − (r − h) × √(2rh − h²).

A minor segment is the smaller region cut off by a chord — corresponding to a central angle less than 180°. A major segment is the larger region on the other side of the chord, with a central angle greater than 180°. Most segment-area formulas target the minor segment by default.

The segment height, also called the sagitta, is the perpendicular distance from the midpoint of the chord to the arc. It is the 'depth' of the segment. Given radius r and central angle θ, the height is h = r × (1 − cos(θ / 2)).

A chord is a straight line segment whose two endpoints both lie on the circle's circumference. It divides the circle into two arcs and two segments. The longest possible chord is the diameter, which passes through the center.

The arc length s of a circular segment equals the radius multiplied by the central angle in radians: s = r × θ. If your angle is in degrees, convert first: θ (radians) = θ (degrees) × π / 180.

Segment area calculations appear in engineering (cross-sections of pipes or tanks), architecture (arched windows and domes), landscaping (curved garden beds), and manufacturing (shaped cuts in materials). Any time a curved boundary meets a straight edge, segment geometry is involved.

Using A = r² × (θ − sin θ) / 2 with r = 5 cm and θ = π/2 radians: A = 25 × (1.5708 − 1) / 2 ≈ 25 × 0.2854 ≈ 7.135 cm². That is the area of the minor segment cut by a 90° central angle in a circle of radius 5 cm.