# Radians to Degrees — Conversion Table and Formula

TL;DR

To convert radians to degrees, multiply by 180°/π. The formula: degrees = radians × 180°/π. This article gives a complete conversion table for every common angle (multiples of π/12, π/6, π/4, π/3, π/2 and beyond), three worked examples, the reverse direction, and the most common mistakes.

## The Conversion Formula

To convert radians to degrees, multiply by 180°/π:

degrees = radians × 180°/π

To convert degrees to radians, multiply by π/180°:

radians = degrees × π/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 2π in radians. So 2π radians = 360°, which simplifies to π radians = 180°, and dividing by π gives 1 radian = 180°/π ≈ 57.2958°.

## The Complete Conversion Table

| Radians          | Degrees | Decimal radians |
| ---------------- | ------- | ---------------- |
| 0                | 0°     | 0.0000          |
| π/12             | 15°    | 0.262           |
| π/6              | 30°    | 0.524           |
| π/4              | 45°    | 0.785           |
| π/3              | 60°    | 1.047           |
| 5π/12           | 75°    | 1.309           |
| π/2              | 90°    | 1.571           |
| 7π/12           | 105°   | 1.833           |
| 2π/3             | 120°   | 2.094           |
| 3π/4             | 135°   | 2.356           |
| 5π/6             | 150°   | 2.618           |
| 11π/12          | 165°   | 2.880           |
| π                | 180°   | 3.142           |
| 7π/6             | 210°   | 3.665           |
| 5π/4             | 225°   | 3.927           |
| 4π/3             | 240°   | 4.189           |
| 3π/2             | 270°   | 4.712           |
| 5π/3             | 300°   | 5.236           |
| 7π/4             | 315°   | 5.498           |
| 11π/6           | 330°   | 5.760           |
| 2π                | 360°   | 6.283           |

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

## Three Worked Examples, From Quick to Stretch

**Quick.** Convert π/4 radians to degrees.

π/4 × 180°/π = 45°

The π in the numerator cancels with the π in the radian value, leaving a clean number.

**Standard (Wrong path first).** Convert 5π/6 radians to degrees.

Wrong path. A student plugs into a calculator without simplifying first: 5 × 3.14159 × 180/3.14159. Two of the π's should cancel exactly, but on a calculator with rounded π, the cancellation introduces a small error — the answer comes out to 149.999...° instead of the exact 150°.

Diagnosing the inefficiency. The π in the radian expression cancels with the π in the formula's denominator. Doing the algebra first gives an exact answer.

Correct path:

5π/6 × 180°/π = 150°

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

**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 π in it, so no cancellation is possible. Apply the formula numerically:

0.45 × 180°/π ≈ 25.8°.

So a 0.45-radian arc is about a 26° swing.

## Why 180°/π? — The Conversion's Origin

The conversion factor traces to one fact: **the arc length of a complete circle equals 2πr** (the formula C = 2πr 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. The full circle is also 360° in degree measure. So 2π radians = 360° — and dividing both sides by 2π gives the equivalence:

1 radian = 180°/π ≈ 57.2958°.

This is why every radian-to-degree formula has 180°/π in it.

## Where Radians and Degrees Are Each Used

- **Degrees** — everyday navigation, construction, school geometry, GPS.
- **Radians** — calculus, physics, wave equations, computer graphics, astronomy.

## The Mistakes Worth Catching

1. **Forgetting to multiply by 180°/π (or by π/180° in reverse).**

2. **Using the wrong direction.** Going from radians to degrees multiplies by 180°/π.

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

## 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.

## Conclusion

To convert radians to degrees, multiply by 180°/π. To go the other way, multiply by π/180°.

## Five Minutes of Practice

1. Convert π/6 radians to degrees.
2. Convert 3π/4 radians to degrees.
3. Convert 1.5 radians to degrees, to one decimal place.
4. Convert 270° to radians (as a multiple of π).
5. Convert −60° to radians.