Relation between radian and real numbers

An angle can be measured two ways: in degrees, where a full turn is chopped into $360$ arbitrary pieces, or in radians, where the measure is defined directly from the geometry of a circle — the ratio of arc length to radius. That ratio is a pure number, with no leftover unit, and this is precisely what lets an angle be treated as an ordinary real number $\theta$ rather than a rotation label. Once that identification is made, every real number names some angle, and conversely, which is exactly what is needed a few pages later to define $\sin x$ , $\cos x$ and the other trigonometric functions for any real $x$ , not just for angles drawn in a diagram. This article builds the degree-radian correspondence from the definition of one radian, derives the master conversion $\pi$ radians $= 180^\circ$ , and applies the arc-length relation $s = r\theta$ to two classic problem types.

The core insight

Radian measure is the ratio of arc length to radius, a dimensionless real number, so $\pi$ radians $= 180^\circ$ converts between degrees and radians and lets trigonometric functions take any real number as input.

The core idea

Degree measure recap. A rotation of $\dfrac{1}{360}$ of a full revolution is called one degree, $1^\circ$ , with $1^\circ = 60'$ (minutes) and $1' = 60''$ (seconds). This system is arbitrary — the number $360$ has no geometric necessity behind it, only historical convention.

Radian measure, defined geometrically. Take a circle of radius $r$ . The angle subtended at the centre by an arc whose length exactly equals the radius $r$ is defined as one radian. Because it is built from the ratio $\dfrac{\text{arc length}}{\text{radius}}$ , radian measure carries no physical unit — length divided by length cancels — so an angle of $\theta$ radians simply is the real number $\theta$ . That is the "relation between radian and real numbers" the topic name refers to: every angle in radians corresponds to exactly one real number, and every real number corresponds to exactly one angle.

The general arc relation. For a circle of radius $r$ , an arc of length $s$ subtends an angle $\theta$ at the centre, related by

\theta = \frac{s}{r}, \qquad \text{equivalently} \qquad s = r\theta,

where: $\theta$ is measured in radians, $s$ and $r$ share the same length unit (so the ratio is dimensionless), and this formula is only valid when $\theta$ is in radians — not degrees. Since a full circle has circumference $2\pi r$ , one complete revolution corresponds to radian measure $\dfrac{2\pi r}{r} = 2\pi$ , and a complete revolution is also $360^\circ$ by definition. Equating the two descriptions of "one full turn" gives the master relation used throughout trigonometry:

2\pi \text{ radians} = 360^\circ \quad\Longrightarrow\quad \pi \text{ radians} = 180^\circ.

Figure

A circle showing an arc of length equal to the radius subtending one radian at the centre, alongside a note that a full revolution equals 2 pi radians and 360 degrees

Because radian measure is just a real number, the word "radian" is often dropped entirely: writing "an angle of $\dfrac{\pi}{4}$ " is understood to mean $\dfrac{\pi}{4}$ radians. This real-number correspondence is exactly what allows the unit-circle definitions of $\sin x$ , $\cos x$ , $\tan x$ to take any real number $x$ as their domain, not merely angles labelled in degrees.

Why the conversion factor is what it is

Derivation AFrom one radian to the degree-radian conversion factors

The goal is to derive, starting only from the definition of one radian as the angle subtended by an arc equal to the radius, the exact conversion factors between degree measure and radian measure — the pair $1$ radian $= \dfrac{180^\circ}{\pi}$ and $1^\circ = \dfrac{\pi}{180}$ radians — using nothing beyond the circumference formula and the definition of degree measure as $\dfrac{1}{360}$ of a revolution.

Step 1. Write the radian measure of a full revolution. For a circle of radius $r$ , the arc length of the entire circle is the circumference $s = 2\pi r$ . Using $\theta = \dfrac{s}{r}$ , the radian measure of one full revolution is $\theta = \dfrac{2\pi r}{r} = 2\pi$ .

Step 2. Write the degree measure of the same full revolution. By the definition of degree measure, one complete revolution is exactly $360^\circ$ , independent of the circle's radius.

Step 3. Equate the two descriptions of the same physical rotation. Since $2\pi$ radians and $360^\circ$ both describe one full turn, $2\pi \text{ radians} = 360^\circ$ . Dividing both sides by $2$ gives the compact master relation $\pi \text{ radians} = 180^\circ$ .

Step 4. Solve for each conversion factor. Dividing $\pi \text{ radians} = 180^\circ$ by $\pi$ gives $1 \text{ radian} = \dfrac{180^\circ}{\pi} \approx 57^\circ 16'$ . Dividing instead by $180$ gives $1^\circ = \dfrac{\pi}{180} \text{ radians}$ . So to convert degrees to radians, multiply by $\dfrac{\pi}{180}$ ; to convert radians to degrees, multiply by $\dfrac{180}{\pi}$ .

Worked example

Problem. Convert $75^\circ$ to radian measure, and convert $\dfrac{5\pi}{6}$ radians to degree measure.

Step 1. To convert $75^\circ$ to radians, multiply by $\dfrac{\pi}{180}$ :

75^\circ = 75 \times \frac{\pi}{180} = \frac{75\pi}{180} = \frac{5\pi}{12} \text{ radians}.

Step 2. To convert $\dfrac{5\pi}{6}$ radians to degrees, multiply by $\dfrac{180}{\pi}$ :

\frac{5\pi}{6} \text{ radians} = \frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = 150^\circ.

So $75^\circ = \dfrac{5\pi}{12}$ radians, and $\dfrac{5\pi}{6}$ radians $= 150^\circ$ .

Competitive stretch

Problem. The minute hand of a clock is $21$ cm long. Find the length of the arc traced by its tip in $20$ minutes. (Use $\pi = \dfrac{22}{7}$ .)

Step 1. In $60$ minutes the minute hand completes one full revolution, an angle of $2\pi$ radians; in $20$ minutes the angle turned is proportional: $\theta = \dfrac{20}{60} \times 2\pi = \dfrac{2\pi}{3}$ radians.

Step 2. Apply $s = r\theta$ with $r = 21$ cm and $\pi = \dfrac{22}{7}$ :

s = 21 \times \frac{2\pi}{3} = 14\pi = 14 \times \frac{22}{7} = 44 \text{ cm}.

The tip of the minute hand traces an arc of $44$ cm in $20$ minutes.

Common traps

Using $s = r\theta$ with $\theta$ still in degrees. The formula is derived from $\theta = s/r$ with $\theta$ in radians only; plugging in a degree value gives an answer far too large.
Mixing up which factor goes which way. Remember "degrees $\times \dfrac{\pi}{180}$ = radians" and "radians $\times \dfrac{180}{\pi}$ = degrees" — swapping the two fractions is the single most common slip.
Rounding $\pi$ too early. Unless a problem explicitly gives an approximation like $\pi = \dfrac{22}{7}$ , leave answers in exact terms of $\pi$ rather than converting to a decimal.
Thinking radian measure needs a written unit. " $\dfrac{\pi}{3}$ " with no degree sign and no "rad" label is, by convention, in radians — this is precisely why radian measure behaves like a plain real number.

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Why does radian measure not need a unit, while degree measure does

Radian measure is defined as a ratio of two lengths, arc length divided by radius, so its units cancel and it is a pure number. Degree measure, by contrast, is defined as a fraction of a full revolution ( $\dfrac{1}{360}$ of a turn) and is conventionally written with the degree symbol to distinguish it. Because radians carry no leftover unit, an angle of $\theta$ radians can be treated exactly like the real number $\theta$ , which is why trigonometric functions are defined for real-number inputs using radian measure.

Do I always have to convert to radians before using s equals r theta

Yes. The relation $s = r\theta$ comes directly from the definition $\theta = s/r$ , and that definition only holds when $\theta$ is measured in radians. If an angle is given in degrees, convert it first by multiplying by $\dfrac{\pi}{180}$ , then apply $s = r\theta$ . Skipping this step is the most common source of wrong answers in arc-length problems.

Is pi radians equal to 180 degrees always true, or does it depend on the circle

It is always true, for any circle, because it comes from comparing two different descriptions of one full revolution and the radius cancels out of the radian count. A full turn is $2\pi$ radians (since the circumference $2\pi r$ divided by $r$ gives $2\pi$ , regardless of what $r$ is) and is also $360^\circ$ by definition. Equating them gives $\pi$ radians $= 180^\circ$ independent of the circle's size.