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

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.