How this calculator works
An arc is a piece of a circle between two points on the edge. Its length is the central angle measured in radians multiplied by the radius, because one radian of turn marks off one radius of arc. The straight chord between the same two points is shorter than the arc except when the arc is tiny. The chord comes from splitting the isosceles triangle at the centre into two right triangles: half the chord is r sin(angle/2). The sector area is the fraction of the full circle occupied by the angle, so it is angle divided by 2pi, times pi r squared. The sector perimeter adds the arc and the two radii. Angles are converted from degrees or radians before calculating.
Formula
Arc length L = rθ, where θ is in radians. Chord c = 2r sin(θ/2). Sector area A = θr²/2. Sector perimeter P = L + 2r. Reversing them gives θ = L/r, r = L/θ, and θ = 2 arcsin(c/(2r)).
Worked example
A central angle of 90 degrees is π/2 radians. With a radius of 5 m, the arc is 5 × π/2 ≈ 7.854 m. The chord is 2 × 5 × sin(π/4) ≈ 7.071 m, the sector area is (π/2) × 25 / 2 ≈ 19.635 m², and the sector perimeter is about 17.854 m. A chord of 5 m in a circle of radius 5 m corresponds to a 60 degree angle and an arc of about 5.236 m.
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Frequently asked questions
Why must the angle be in radians in the formula?
A radian is defined so that arc length equals radius times angle. Degrees measure the same turn with a different unit, so multiplying by degrees would give the wrong length. The calculator converts your angle to radians first, whichever unit you choose.
How is the chord different from the arc?
The arc follows the curve and the chord cuts straight across. For a 90 degree arc, the chord is about 90 percent of the arc length. For a very small angle they are almost equal, and for a half circle the chord is the diameter while the arc is half the circumference.
Can the central angle be larger than half a turn?
Yes, up to a full turn. A reflex arc is more than half the circle. The chord formula still works because the sine of an angle above π/2 is symmetric around the half turn, and the sector area grows with the angle.
What if the chord is longer than the diameter?
That cannot happen in a circle, so the calculator rejects it. The longest possible chord is the diameter, which corresponds to a half-turn arc. Check that the radius and chord use the same unit before entering them.