Using the calculator
Type the radius and the central angle, like r = 6, θ = 120°, or two plain numbers, 6, 120. The angle is read in degrees unless you write rad after it, use π (2π/3), or set "Angles in" to Radians.
The arc length is the answer at the top, exact where it can be (4π cm here) with the decimal beside it. Under it are the sector area, the chord, the segment and the sector's perimeter, and the drawing shades the sector and draws the arc heavy.
If you know the diameter or the circumference instead of the radius, use d = or C = in place of r =. To find the angle from an arc you've measured, type the radius and the arc, as in r = 6, s = 10.
The formula
With the angle in radians, s = rθ. With it in degrees, s = 2πr × θ/360.
The degrees version says what's going on. The whole way round is 2πr, and an angle of θ degrees is θ/360 of the whole way round. 90° is a quarter, so the arc is a quarter of the circumference.
Radians make it shorter because a radian is defined by arc length. One radian is the angle at the centre when the arc is as long as the radius, so θ radians of arc is θ radii long. Converting is × π/180 one way and × 180/π the other; the degrees to radians converter shows the working.
Arc length for common angles
| Angle | Radians | Fraction of the circle | Arc length | Sector area |
|---|---|---|---|---|
| 30° | π/6 | 1/12 | πr/6 | πr²/12 |
| 45° | π/4 | 1/8 | πr/4 | πr²/8 |
| 60° | π/3 | 1/6 | πr/3 | πr²/6 |
| 90° | π/2 | 1/4 | πr/2 | πr²/4 |
| 120° | 2π/3 | 1/3 | 2πr/3 | πr²/3 |
| 180° | π | 1/2 | πr | πr²/2 |
| 270° | 3π/2 | 3/4 | 3πr/2 | 3πr²/4 |
| 360° | 2π | 1 | 2πr | πr² |
Worked examples
A clock's minute hand 9 cm long, over 20 minutes. Twenty minutes is a third of an hour, so 120°. The tip travels s = 2π × 9 × 120/360 = 6π cm, about 18.85 cm.
An angle in radians. A radius of 2.5 m and an angle of 1.2 rad give s = 2.5 × 1.2 = 3 m, no π involved.
From the arc to the angle. A curved path 10 m long on a circle of radius 8 m spans θ = s/r = 1.25 rad, about 71.62°.
Arc against chord. On a circle of radius 10, a 60° arc is 10π/3, about 10.47, but the straight chord across it is exactly 10, since the triangle from the centre is equilateral. The gap grows with the angle: at 180° the arc is 10π, about 31.42, and the chord is the diameter, 20.
Who named the radian
Roger Cotes, who proofread the second edition of Newton's Principia, treated an angle as an arc length in his 1714 paper Logometria. After he died, his cousin Robert Smith published his collected papers in 1722, and with them what is probably the first calculation of one radian in degrees, about 57.2958°.
The name came more than 150 years later. It first appears in print on 5 June 1873, in exam questions James Thomson set at Queen's College, Belfast. Thomas Muir at St Andrews had been trying out rad, radial and radian, and settled on radian in 1874 after talking it over with Thomson.
Questions
What's the difference between arc length and chord length?
The arc follows the curve; the chord is the straight line between its ends. The chord is always shorter, c = 2r sin(θ/2), and both are listed under the answer.
How do I find the arc length without the angle?
You need one more measurement besides the radius. With the sector area K, the angle is θ = 2K/r², and from that s = rθ. Type r = 6, K = 30 to try it.
Can I use the diameter?
Yes. Type d = 12, θ = 120° and the calculator halves it for you.
Sources
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