🌀Radians & Degrees

Degrees are a human invention: somebody decided that a full turn would be cut into 360 equal pieces, probably because 360 has so many convenient factors. There is nothing mathematical about the number.

Radians measure an angle by the arc it cuts. Stand at the centre of a circle of radius r and open an angle until the arc between the arms is exactly r units long. That angle is 1 radian, and it is the same angle in every circle.

Since the whole circumference is 2 pi r, a full turn is 2 pi radians. Therefore

360° = 2 pi radians, so 180° = pi radians

Everything else follows from that one line. Radians are not an alternative notation for degrees; they are a genuinely better unit, and the calculus formulas for sine and cosine only work in radians.

From the fact that 180° = pi radians:

degrees -> radians: multiply by pi/180 radians -> degrees: multiply by 180/pi

60° = 60 x pi/180 = pi/3 radians 3*pi/4 radians = (3 pi/4) x 180/pi = 135°

Two habits make this reliable. First, simplify the fraction rather than reaching for a decimal: 60/180 = 1/3, so the answer is pi/3, exactly. Second, sanity-check the size. A radian is a bit under 60°, so a radian measure is always a much smaller number than the same angle in degrees. If a conversion makes the number bigger, it went the wrong way.

Angles worth knowing by heart:

30° = pi/6 45° = pi/4 60° = pi/3 90° = pi/2 120° = 2*pi/3 135° = 3*pi/4 180° = pi 360° = 2*pi

Radians pay for themselves immediately. The length of an arc cut by a central angle t is

s = r x t (t in radians)

That is it — no fractions of 360, no extra constants. It is the definition of a radian written as a formula.

A circle of radius 8 with a central angle of 3*pi/4 radians has arc length 8 x 3*pi/4 = 6*pi.

If the angle arrives in degrees, convert it first. Using degrees directly in s = r t gives an answer that is wrong by a factor of about 57.

Working in degrees the arc length formula has to carry the fraction of the circle instead: s = (t/360) x 2 pi r. Both give the same answer; the radian version is simply shorter.

The area of a sector is just as tidy in radians:

A = (1/2) r² t (t in radians)

For radius 6 and angle pi/3: A = (1/2)(36)(pi/3) = 6*pi.

Keep the two formulas apart the way you would for circles. An arc is a length and uses r once; a sector is an area and uses r squared. And do not lose the 1/2 — without it you have the area of a shape twice the size.

It is also worth being able to place a radian angle on the plane without converting. The quadrant boundaries are pi/2, pi and 3*pi/2, so an angle of 5*pi/6 sits between pi/2 and pi and is therefore in the second quadrant. Comparing against those three landmarks is quicker than converting to degrees every time.

Attach a clock to an angle and it becomes a speed. Angular speed is the angle swept per unit of time:

angular speed = angle / time

The only trap is the unit of the angle. One revolution is 2*pi radians, so a wheel turning 3 times in 2 seconds has angular speed 3 x 2*pi / 2 = 3*pi radians per second — not 1.5, which counts turns rather than radians.

Revolutions per minute (rpm) is the everyday unit, and reaching radians per second from it takes two conversions: multiply by 2*pi to turn revolutions into radians, then divide by 60 to turn minutes into seconds.

45 rpm = 45 x 2*pi / 60 = 4.71 radians per second

Linear speed is how fast a point on the rim actually travels along its arc. Since arc length is s = r t, dividing both sides by the time gives

v = r x angular speed

and again only in radians — it is s = r t wearing a stopwatch. A point 12 cm from the centre of that turntable moves at 12 x 4.71 = 56.5 cm/s, while a point half as far out moves half as fast through the same angle.

Run the chain backwards and a road speed becomes revolutions. Every turn of a wheel carries a bicycle forward by exactly one circumference, 2 pi r — or pi d, which is the same length, so use one or the other and never both. Divide the distance covered in a minute by that circumference and the answer is rpm.