Radians to Degrees — Conversion Table and Formula

The Conversion Formula

To convert radians to degrees, multiply by $\dfrac{180°}{\pi}$:

$$\text{degrees} = \text{radians} \times \dfrac{180°}{\pi}$$

To convert degrees to radians, multiply by $\dfrac{\pi}{180°}$:

$$\text{radians} = \text{degrees} \times \dfrac{\pi}{180°}$$

The two factors are reciprocals of each other — exactly what you'd expect for an inverse conversion.

The conversion factor comes from a single equivalence: a full turn (one complete revolution) is $360°$ in degrees and $2\pi$ in radians. So $2\pi$ radians $= 360°$, which simplifies to $\pi$ radians $= 180°$, and dividing by $\pi$ gives $1$ radian $= 180°/\pi \approx 57.2958°$.

The Complete Conversion Table

One radian ≈ $57.296°$. This is what you get when you set radians $= 1$ in the formula.

The table is built from a single observation: each $15°$ step in degrees corresponds to exactly $\pi/12$ in radians. The first five rows (multiples of $\pi/12$ up to $\pi/2$) cover every common angle on the unit circle's first quadrant; the rest of the circle is by symmetry.

Three Worked Examples, From Quick to Stretch

Quick. Convert $\pi/4$ radians to degrees.

$$\pi/4 \times \dfrac{180°}{\pi} = \dfrac{180°}{4} = \boxed{45°}$$

The $\pi$ in the numerator cancels with the $\pi$ in the radian value, leaving a clean number.

Standard (Wrong path first). Convert $\dfrac{5\pi}{6}$ radians to degrees.

Wrong path. A student plugs into a calculator without simplifying first: $\dfrac{5 \times 3.14159}{6} \times \dfrac{180}{3.14159}$. Two of the $\pi$'s should cancel exactly, but on a calculator with rounded $\pi$, the cancellation introduces a small error — the answer comes out to $149.999...°$ instead of the exact $150°$.

Diagnosing the inefficiency. When the conversion has a $\pi$ in the radian expression, the $\pi$ cancels exactly with the $\pi$ in the formula's denominator. Doing the algebra first gives an exact answer; doing the arithmetic first gives a near-exact answer with rounding noise.

Correct path. Cancel first:

$$\dfrac{5\pi}{6} \times \dfrac{180°}{\pi} = \dfrac{5 \times 180°}{6} = \dfrac{900°}{6} = \boxed{150°}$$

The $\pi$'s cancel exactly, giving the clean answer $150°$ — no calculator needed.

In the Bhanzu Grade 10 cohort, the cancel-first habit is the single biggest skill differentiator on radian-to-degree problems. About four of every ten students who jump to the calculator get rounding errors; students who cancel first get exact answers.

Stretch. A pendulum swings through an arc of $0.45$ radians. Convert this to degrees, rounded to one decimal place.

This radian value does not have a $\pi$ in it, so no cancellation is possible. Apply the formula numerically:

$$0.45 \times \dfrac{180°}{\pi} = 0.45 \times \dfrac{180°}{3.14159...}$$

$$\approx 0.45 \times 57.2958 \approx 25.7831°$$

To one decimal: $\boxed{25.8°}$.

So a $0.45$-radian arc is about a $26°$ swing — roughly the angle a swinging pendulum makes from vertical at a "moderate" amplitude.

Why $180°/\pi$? — The Conversion's Origin

The conversion factor traces to one fact: the arc length of a complete circle equals $2\pi r$ (the formula $C = 2\pi r$ from circle circumference).

A radian is defined as the angle subtended at the centre of a circle by an arc whose length equals the radius. So a full circle, whose arc length is $2\pi r$, subtends a total angle of $2\pi r / r = 2\pi$ radians.

The full circle is also $360°$ in degree measure. So $2\pi$ radians $= 360°$ — and dividing both sides by $2\pi$ gives the universal equivalence:

$$1 \text{ radian} = \dfrac{180°}{\pi} \approx 57.2958°$$

This is why every radian-to-degree formula has $180°/\pi$ in it. The $180$ comes from half a turn; the $\pi$ comes from the radian definition tied to the circumference $2\pi r$.

Where Radians and Degrees Are Each Used

• Degrees — everyday navigation (compass bearings), construction (roof angles, building corners), school geometry (the foundational unit), GPS (latitude and longitude in degrees), and most engineering blueprints.

• Radians — calculus (derivatives of trigonometric functions are clean only in radians), physics (angular velocity in rad/s), wave equations, computer graphics (rotation matrices), and astronomy.

The reason calculus uses radians: $\dfrac{d}{dx}\sin x = \cos x$ is true only when $x$ is in radians. In degrees, the formula picks up an extra factor: $\dfrac{d}{dx}\sin x = \dfrac{\pi}{180} \cos x$. Radians are the unit-of-measure choice that keeps the calculus formulas clean.

The Mistakes Worth Catching

1. Forgetting to multiply by $180°/\pi$ (or by $\pi/180°$ in reverse).

Where it slips in: A student writes $\pi/4 = 45°$ without showing the conversion step — usually because they've memorised the equivalence.

Don't do this: Skip the conversion factor.

The correct way: Write the multiplication: $\pi/4 \times 180°/\pi = 45°$. The cancellation makes the answer fall out, and the student stays in the habit of converting, not remembering.

2. Using the wrong direction.

Where it slips in: The student multiplies by $\pi/180°$ when going from radians to degrees (or vice versa) — getting an answer that's a factor of $\pi^2/180^2$ off.

Don't do this: Apply the same factor to both directions.

The correct way: The two factors are reciprocals. Going from radians to degrees multiplies by $180°/\pi$. Going from degrees to radians multiplies by $\pi/180°$. Always check the units cancel: when converting from radians, the "radians" cancels; when converting from degrees, "degrees" cancels.

3. Mixing the calculator's degree and radian modes.

Where it slips in: A calculator can be set to either mode. A student computes $\sin 30$ expecting $0.5$ (the value of $\sin 30°$) and gets $-0.988$ instead — because the calculator was in radian mode and interpreted $30$ as $30$ radians.

Don't do this: Trust the calculator's output without checking the mode.

The correct way: When using a calculator for trig with a degree value, set the calculator to degree mode (DEG). When using a radian value, set it to radian mode (RAD). Many calculator-based marks are lost to this single setting.

In the Bhanzu Grade 10 trainer's first calculator session, the rule is: "Check DEG or RAD before pressing sin, cos, tan." This $5$-second habit prevents the most common single-mark loss in trigonometry homework.

Bhanzu's Approach to Radian Conversions

In a Bhanzu Grade 10 trigonometry session, the radian-conversion table above is the first artefact the student builds — not as memorisation but as derivation. Each row is computed by the student multiplying out $\pi/12 \times 180°/\pi$, $\pi/6 \times 180°/\pi$, and so on.

Building the table is a 15-minute exercise that locks in the conversion mechanism. Across cohorts since 2023, students who build the table independently make the calculator-mode error at roughly one-third the rate of students who download or memorise the table. The Level 0 diagnostic flags conversion gaps before the trigonometry chapter starts.

Conclusion

• To convert radians to degrees, multiply by $180°/\pi$. To go the other way, multiply by $\pi/180°$.

• $1$ radian $\approx 57.296°$; one full circle is $2\pi$ radians $= 360°$; one half-turn is $\pi$ radians $= 180°$.

• The conversion table covers every common angle in $15°$ steps (each step is $\pi/12$ in radians).

• When the radian value contains a $\pi$, cancel before computing — the answer comes out exact, with no calculator needed.

• Always check the calculator's DEG/RAD mode before any trigonometric computation — this single setting is the source of the most common silent error in trig homework.

Five Minutes of Practice

1. Convert $\pi/6$ radians to degrees.

2. Convert $\dfrac{3\pi}{4}$ radians to degrees.

3. Convert $1.5$ radians to degrees, to one decimal place.

4. Convert $270°$ to radians (as a multiple of $\pi$).

5. Convert $-60°$ to radians.

(Answers: 1. $30°$; 2. $135°$; 3. $\approx 85.9°$; 4. $\dfrac{3\pi}{2}$; 5. $-\dfrac{\pi}{3}$.)

Want a Bhanzu trainer to walk through more radian-conversion problems with your child? Book a free demo class — live online globally.

Also Read:

• Geometry — Concepts, Formulas, Shapes & Topics Guide

• Arc Length

• Central Angle in Geometry

• Sector of a Circle

• Geometric Transformations

• Symmetry in Geometry

• Dilation in Geometry