Radians

A radian is the angle at the center of a circle that cuts off an arc exactly as long as the radius. Take the radius, bend it around the rim, and the angle it spans is one radian, about 57.3°. The whole rim is 2π ≈ 6.283 radii long, so a full turn is 2π radians, a half turn is π, and a right angle is π/2^[1].

Degrees divide a turn into 360 parts because someone chose 360. Radians divide it by a length the circle already has. That one difference is why every formula in calculus, physics and engineering that involves an angle assumes the angle is in radians.

A radius, bent around the rim§

Notice two things. Six radii fit, with a little over a quarter of a radius to spare: the rim is 2π radii long, and 2π = 6.28318… Second, when you change r, the arc grows or shrinks with it, but the number of radii in the arc does not. That number is the angle in radians. In symbols, an arc of length s on a circle of radius r subtends an angle

On the unit circle r = 1, so the angle in radians is the arc length. That is the deepest reason for drawing the circle with radius 1: the angle you turn through and the distance your point walks along the rim become the same number.

Converting degrees and radians§

Half a turn is 180° and also π radians, which gives every conversion:

To turn degrees into radians, multiply by π/180; to go back, multiply by 180/π. So 30° is 30π/180 = π/6, 45° is π/4, 60° is π/3, and 270° is 3π/2. Writing the answer as a multiple of π keeps it exact; 0.5236 is only an approximation to π/6.

One confusion is worth naming because it is common even among people who teach the subject. “π = 180°” is a useful shorthand, and it is also false: π is a number, 3.14159…, and it is π radians that equals 180°. In a study of 42 trainee mathematics teachers, Hatice Akkoç found that their idea of a radian was dominated by their idea of a degree, that many held two separate images of π (an angle and an irrational number) without connecting them, and that some were reluctant to accept a trigonometric function of a real number such as sin 2^[2]. documented The bent radius is the cure: an angle of 2 radians is the angle under an arc two radii long, nothing more mysterious, and sin 2 is the height of the point you reach.

Why mathematicians insist on radians§

For drawing a pie chart, degrees are fine. The reason radians win everywhere else is a single fact about small angles. Look at a thin slice of the unit circle near (1, 0). The height of the point (sin θ), the arc (θ itself, since the radius is 1) and the yellow segment on the tangent line (tan θ) are three lengths that are nearly the same.

The picture proves an inequality you can read off it directly: for 0 < θ < π/2,

Divide through by sin θ and you get cos θ < sin θ / θ < 1. As θ shrinks, cos θ goes to 1, so sin θ / θ is squeezed to 1 as well^[3]. That limit is the foundation of the calculus of trigonometric functions: it is why the derivative of sin x is cos x. But it holds only because θ on the middle of the inequality is an arc length, which means the angle is in radians. Measure in degrees and the ratio tends to π/180 ≈ 0.01745, and every derivative drags that factor along: the slope of sin(x°) is (π/180) cos(x°).

A unit that is really the number 1§

Since an angle in radians is a length divided by a length, it has no physical dimension. The International System of Units (SI) says so explicitly: the radian is a derived unit, rad = m/m = 1^[5]. established Writing “rad” is a courtesy that tells the reader how the number was measured, and you may drop it. That is why sin 2 makes sense as written.

Not every metrologist is comfortable with this. The radian was classed as a “supplementary unit” until 1995, when it became a dimensionless derived unit, and the choice still has critics. Peter Quincey of the UK’s National Physical Laboratory has set out the range of alternatives, including making angle a dimension of its own, and concluded that the current treatment, though less strictly logical than some options, is broadly supported^[6]. The BIPM revised its description of angle units in the 2024 edition of the SI Brochure without changing their status^[5]. active

3438: a radian, in arcminutes§

Long before anyone said “radian”, astronomers were using one without naming it. Divide a circle into 360 degrees of 60 arcminutes each and its rim has 21,600 arcminutes. If you measure the radius in the same unit as the rim, it comes out at 21,600 / 2π ≈ 3437.75 arcminutes. Round that and you get 3438.

That is the radius Āryabhaṭa used for his sine table in 499. In a single verse of the Āryabhaṭīya he lists 24 differences, 225, 224, 222, 219, 215, … , 22, 7, in arcminutes. Add them up one by one and you get his sines, half-chords at steps of 3°45′ in a circle of radius 3438^[7][8]. documented All 24 differences add up to exactly 3438, the half-chord of a quarter circle, which is the radius itself.

The engine behind this figure rebuilds the table from his differences and compares it with the modern values: every one of the 24 entries is within 0.71 of an arcminute (the worst is at 67°30′), and six are within a tenth. A radius of 3438 is why the first entry works so neatly: for a small angle the half-chord is almost the arc, and the arc of 3°45′ is 225 arcminutes long, so the first sine is 225. In modern terms he had chosen a circle in which the sine of a small angle is nearly the angle in the rim’s own units, which is the property that makes radians natural. Toomer argued that Hipparchus’s lost chord table used the same radius, which would push the idea back six centuries more^[9]; the argument is disputed^[10]. contested

Who named the radian§

The word is much younger than the idea. According to Florian Cajori’s history of 1919, it first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queen’s College, Belfast. The Scottish mathematician Thomas Muir had been hesitating since 1869 between “rads”, “radials” and “radians”, and adopted “radians” in 1874 after consulting Thomson^[11]. documented The same collection of sources also quotes Thomson and Tait’s Treatise on Natural Philosophy calling the unit angle “a radian”, under the date 1867, so the 1873 date may not be the first. The measure itself is older than the word: Roger Cotes is credited with radian measure early in the eighteenth century^[12], and in Euler’s Introductio of 1748 the radius is 1, so the arc and the angle are the same number^[13].

Radians are the unit in which a point going around a circle at constant speed has coordinates that are the simplest possible functions of time. Chapter 3 unrolls those functions into waves.