How this calculator works
A chord is the straight line joining two points on a circle. Together with the two radii drawn to its ends it forms an isosceles triangle, which is why half the chord, the apothem and the radius make a right triangle. The central angle fixes how much of the circle the chord cuts off. The chord height is the distance from the middle of the chord to the arc, the apothem is the remaining distance to the centre, and the arc length and two areas follow from the angle in radians. The chord and height modes run the same geometry in reverse.
Formula
c = 2r sin(θ/2); h = r(1 - cos(θ/2)); apothem = r cos(θ/2); arc = rθ; sector area = ½r²θ; segment area = ½r²(θ - sin θ); r = (c²/4 + h²) / (2h). The angle θ is in radians.
Worked example
A radius of 5 m and a central angle of 60° give c = 10 sin 30° = 5 m, h = 5(1 - cos 30°) ≈ 0.6699 m, and a segment area of ½ × 25 × (π/3 - sin 60°) ≈ 2.2647 m².
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Frequently asked questions
Is the chord the same as the arc?
No. The chord is the straight path between the two points, while the arc follows the circle. The arc is always longer than the chord for the same pair of endpoints.
What is the chord height?
Also called the sagitta, it is the greatest distance from the chord to the arc. It is measured perpendicular to the chord at its midpoint.
What if the central angle is 180°?
The chord is then a diameter and the height equals the radius. A 360° angle returns a chord of zero and a full circle, which is the limit of the calculation.
Why can the chord not exceed the diameter?
The longest possible straight line inside a circle passes through the centre and is the diameter. A longer chord cannot fit in a circle of that radius.