How this calculator works
A circular segment is the region between a chord and the arc it cuts off. Its area is the area of the corresponding sector minus the area of the isosceles triangle formed by the two radii and the chord. Written with the central angle in radians, that difference is ½r²(θ - sin θ). The chord, segment height and arc length all follow from the same angle, so knowing the radius plus any one of them determines the rest. The segment perimeter is the curved arc plus the straight chord.
Formula
Segment area = ½r²(θ - sin θ); chord = 2r sin(θ/2); height = r(1 - cos(θ/2)); arc length = rθ; sector area = ½r²θ; segment perimeter = arc length + chord. The angle θ is in radians.
Worked example
A radius of 5 m with a central angle of 90° gives a segment area of ½ × 25 × (π/2 - 1) ≈ 7.1350 m², a chord of 5√2 ≈ 7.0711 m, a height of 5(1 - cos 45°) ≈ 1.4645 m and a perimeter of about 14.9250 m.
Advertisement
Frequently asked questions
What is the difference between a segment and a sector?
A sector is the wedge bounded by two radii and an arc. A segment is bounded by a chord and an arc. The segment is the sector with the triangle removed.
Why is the angle in radians?
The formula ½r²(θ - sin θ) assumes θ is in radians, because arc length is rθ only in that unit. The input is converted from degrees automatically when you choose degrees.
Can the segment be more than half the circle?
Yes. Angles beyond 180 degrees describe a major segment, and the formula still works. The height can then exceed the radius, approaching the diameter as the angle approaches 360 degrees.
Which height is measured?
The height is the greatest distance from the chord to the arc, measured perpendicular to the chord at its midpoint. It is also called the sagitta.