Why Angles Are Measured in Radians

Set a radius and an angle, then switch the unit underneath. The arc length, the slope of sine and the small-angle approximation move together, because they are one definition seen three ways.

Worked example — Why Angles Are Measured in Radians

Arc length s	1
Slope of sin at θ	0.540302
sin θ	0.841471
Error in sin θ ≈ θ	18.840%

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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The arc formula is the definition, not a result 🖖

Press One radian, one radius. A radius of 1 turned through 1 radian cuts an arc of exactly 1, and that is what the unit means: the angle that makes the arc equal the radius. So s = rθ is not a theorem about radians, it is the sentence radians were invented to make true. Switch the unit to degrees and the same arc needs r × θ × π/180. Drop that factor and the answer is out by 57.3, which is not a random constant either. It is one radian written in degrees, which is the whole conversion staring back at you.

The derivative of sine is cosine only in radians 🖖

Press The slope calculus depends on. At θ = 0 the slope of sine is exactly 1. Now switch the unit: the same point on the same curve reads 0.017453, because a degree is a smaller step than a radian and sine climbs proportionally less on each one. The clean rule d/dx sin x = cos x is therefore a statement about radians and about no other unit. So is the series for sine, so is every Fourier term, and so is the small-angle result below. A degree-based calculus is perfectly possible, and the factor never leaves: a first derivative carries π/180, a second carries its square, an integral carries the reciprocal. That is the reason nobody builds one.

The approximation survives much further than it looks like it should 🖖

For a small angle, sin θ ≈ θ. The next term of the series is −θ³/6, so the relative error grows like θ²/6 — with the square of the angle, not with the angle. That is why the approximation lasts: at 1° it is out by 0.005%, at 5° by 0.127%, and even at 10° by only 0.510%. Push on to 30° and it finally breaks, at 4.720%. This is exactly the licence a pendulum runs on. Its motion is only simple harmonic because sin θ has been replaced by θ, so a pendulum keeps time at small swings and slowly loses it at large ones, and the point at which a clock starts drifting is the point where this column stops being small.

Problems solved in full

Diameter of the Moon covering 0.5181° of sky at 384,400 km away 6 steps
The Moon covers 0.5181° of sky and is 384,400 km away. Work out its diameter, and work out what forgetting the unit would cost you.
1. The formula for a small distant object is size = distance × angle, which is s = rθ with the Moon's distance as the radius. It needs θ in radians.
  
  $s = r θ (θ in radians)$
2. Convert: 0.5181 × π/180 = 0.0090426 radians.
  
  $0.518 1^{\circ} \times \frac{π}{180} = 0.0090426$
3. Multiply. 384,400 × 0.0090426 = 3,476 km, against a measured mean diameter of 3,474.8 km.
  
  $384, 400 \times 0.0090426 = 3, 476 km$
4. Now forget the conversion and multiply by the raw 0.5181. That gives 199,158 km, a Moon most of the way to being a second Earth-Moon distance across.
  
  $384, 400 \times 0.5181 = 199, 158 km$
5. The ratio of the two answers is 57.30, which is 180/π. Every dropped conversion of this kind is out by exactly that, whatever the numbers were.
  
  $\frac{199 , 158}{3 , 476} = 57.30 = \frac{180}{π}$
6. One check before trusting the answer. The formula treats the Moon's diameter as an arc rather than a straight line across it, and at this angle those differ by 12 metres in 3,476 km, or 3 parts in a million. The approximation is not what limits the result.
  
  $arc - chord = 12 m in 3, 476 km$
Answer

3,476 km, and forgetting the radians costs a factor of 57.3. What limits the fourth digit is not the geometry but the pairing. The Moon's distance runs from about 363,300 km to 405,500 km over a month, an 11% swing, and its angular size swings with it. Even a mean angle against a mean distance lands 1.2 km out, which is the gap between the 3,476 above and the measured 3,474.8, because angular size goes as 1/d and the average of a reciprocal is not the reciprocal of the average. Pair that mean angle with tonight's distance instead and you are 190 km out at perigee, with the arithmetic flawless throughout. The equation cannot tell you that the two numbers you fed it came from different moments.
θ is bracketed by sine and tangent, and not down the middle 6 steps
At 10° the panel prints an arc of 0.174533, a sine of 0.173648 and an error of 0.510%. Work out where the tangent of the same angle falls, and decide whether 0.510% is the best those two numbers can do.
1. Ten degrees is 10 × π/180 = 0.174533 radians, and on a circle of radius 1 the arc is the angle. That is the whole content of s = rθ.
  
  $θ = 1 0^{\circ} \times \frac{π}{180} = 0.174533 rad, s = r θ = θ (r = 1)$
2. The sine comes in a little short at 0.173648. The panel divides the shortfall by the sine and reports 0.510%.
  
  $sin θ = 0.173648, \frac{θ - sin θ}{sin θ} = \frac{0.000885}{0.173648} = 0.510%$
3. The shortfall has a size you can predict. sin θ = θ − θ³/6 + …, so θ − sin θ should be about θ³/6 = 0.000886, against a measured 0.000885.
  
  $sin θ = θ - \frac{θ ^{3}}{6} + \dots ⟹ θ - sin θ \approx \frac{θ ^{3}}{6} = 0.000886$
4. The tangent overshoots by the same kind of term with twice the coefficient: tan θ = θ + θ³/3 + …, giving tan θ − θ = 0.001794. Together, sin θ < θ < tan θ. That bracket is the squeeze which proves the slope of sine at zero is exactly 1 in radians, and the page prints only its lower half.
  
  $tan θ = θ + \frac{θ ^{3}}{3} + \dots = 0.176327 ⟹ tan θ - θ = 0.001794 \approx \frac{θ ^{3}}{3}$
5. The bracket is not symmetric. The tangent overshoots 2.03 times as far as the sine falls short, so θ sits one third of the way up from sine to tangent rather than halfway.
  
  $sin θ < θ < tan θ, \frac{tan θ - θ}{θ - sin θ} = 2.03 ⟹ θ lies one third of the way up$
6. Weight them in that ratio and the cubic terms cancel outright, since 2(−⅙) + ⅓ = 0. (2 sin θ + tan θ)/3 = 0.1745411, out by 8.2 × 10⁻⁶, or 0.0047% — more than a hundred times closer than the sine alone, from the same two numbers.
  
  $2 (- \frac{1}{6}) + \frac{1}{3} = 0 : \frac{2 sin θ + tan θ}{3} = 0.1745411, out by 8.2 \times 1 0^{- 6} = 0.0047%$
Answer

The tangent lands at 0.176327, on the far side of the arc, and 0.510% is a long way from the best available. This weighting has a long history under another name: a regular polygon inscribed in a circle has perimeter 2n sin(π/n) and the circumscribed one 2n tan(π/n), the same sine and tangent at θ = π/n, and dividing by the diameter gives the classical bounds on π. At 96 sides those are 3.14103195 and 3.14271460, which on their own settle π to 3.14; combined as (2 × 3.14103195 + 3.14271460)/3 they give 3.14159283 against a true 3.14159265. The tangent has been in the tool's own model all along; only the sine gets a card, because the small-angle rule is what a physics course asks for.

$n = 96 : \frac{2 ( 3.14103195 ) + 3.14271460}{3} = 3.14159283 (π = 3.14159265)$

Learning path

Why everything looks straight up close

Leads to Limits the angle measure in which sin θ ≈ θ holds for small θ and the derivative of sine is exactly cosine.

By the 42Math team · Updated 2026-09-03

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

One radian, one radius - A radius of 1 and an angle of 1 radian cut an arc of exactly 1. That is not a result about radians, it is what the unit was defined to do, and every other formula on this page follows from it.
The slope calculus depends on - At θ = 0 the slope of sine is exactly 1 in radians. Switch the unit to degrees and the same slope reads 0.017453, because sine now rises 57.3 times more slowly per unit of angle. Every derivative you have met assumes the first number.
Where sin θ ≈ θ gives out - At 10° the approximation sin θ ≈ θ is out by 0.510%. At 5° it is 0.127% and at 1° it is 0.005%, because the error grows with the square of the angle rather than with the angle.
The Moon from its angular size - The Moon covers 0.5181° of sky at a mean distance of 384,400 km. In radians that angle is 0.0090426, and multiplying gives a diameter of 3,476 km against a measured 3,474.8. Forget the conversion and the answer is 199,158 km.