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Radian and Degree Conversion
Radian and Degree Conversion: Formulas, Chart & Examples
Quick Answer
To convert between degrees and radians, use two formulas:
Degrees → Radians
degrees × π/180
Radians → Degrees
radians × 180/π
Example: 90° × π/180 = π/2 radians. Example: π/3 × 180/π = 60°.
In This Guide
- What a Radian Actually Is
- Why π/180 Specifically
- The Two Conversion Formulas
- Converting Degrees to Radians
- Converting Radians to Degrees
- Full Reference Table — All 16 Unit Circle Angles
- Recognizing Radian Form Without Converting
- Negative Angles and Angles Beyond 2π
- Calculator Mode: Degrees vs Radians
- The Calculus Connection
- Common Mistakes
- Practice Problems
- Platform Tips
- Frequently Asked Questions
What a Radian Actually Is
Most students learn the conversion formula without ever understanding what a radian means. That’s a problem — when you understand what a radian is, the formula π/180 makes immediate sense and you never have to memorize which direction to convert.
A radian is defined by the relationship between a circle’s radius and its arc length. If you take the radius of any circle and lay it along the circumference as an arc, the angle that arc subtends at the center is exactly 1 radian. The radius and the arc length are equal — that’s it. That’s the entire definition.
Left: when arc length equals radius, the angle is 1 radian (≈ 57.3°). Right: exactly 2π of those arcs fit around the full circle — which is why a full circle equals 2π radians.
Because the circumference of a circle is 2πr, exactly 2π radii fit around the full circle. That means a full rotation (360°) equals 2π radians. Everything else follows from that single fact.
💡 Why This Matters
Radians aren’t arbitrary. They’re the natural unit of angle measurement because they connect angle directly to arc length without any conversion factor. In a unit circle (radius = 1), the arc length of an angle is numerically identical to the angle in radians. That elegant relationship is why mathematicians, physicists, and engineers prefer radians — and why calculus requires them.
Why π/180 Specifically
Students who just memorize the formula are helpless if they blank on an exam. Here’s the three-line derivation that lets you reconstruct it from scratch:
Line 1: A full circle = 360° = 2π radians (from the definition above)
Line 2: Divide both sides by 360: 1° = 2π/360 radians = π/180 radians
Line 3: Divide both sides of Line 1 by 2π instead: 1 radian = 360°/2π = 180°/π
That’s the whole derivation. π/180 converts one degree into its radian equivalent. 180/π converts one radian into its degree equivalent. Multiply by the number of degrees or radians you have and you’re done.
The Two Conversion Formulas
Degrees → Radians
radians = degrees × π/180
Multiply by π, divide by 180
Example: 60° × π/180 = π/3
Radians → Degrees
degrees = radians × 180/π
Multiply by 180, divide by π
Example: π/4 × 180/π = 45°
Memory shortcut: radians involve π — so the formula that produces radians must have π in the numerator (× π/180). The formula that removes radians puts π in the denominator (× 180/π) to cancel it out.
Converting Degrees to Radians
Write the degree value
Start with the degree measure you want to convert. Keep the degree symbol to remind yourself what you’re working with.
Multiply by π/180
Write it as a fraction: (degree value × π) / 180. Keep π in the expression — don’t convert it to 3.14159 yet.
Simplify the fraction
Divide the degree value and 180 by their GCD. Most standard angles simplify cleanly — the answer should be a simple fraction times π.
Worked Examples
| Degrees | × π/180 | Simplify | Result |
|---|---|---|---|
| 30° | 30π/180 | ÷ 30 top and bottom | π/6 |
| 45° | 45π/180 | ÷ 45 top and bottom | π/4 |
| 90° | 90π/180 | ÷ 90 top and bottom | π/2 |
| 135° | 135π/180 | ÷ 45 top and bottom | 3π/4 |
| 270° | 270π/180 | ÷ 90 top and bottom | 3π/2 |
Converting Radians to Degrees
Write the radian value
Identify whether the value is in terms of π (exact form like 3π/4) or a decimal approximation. Exact form is almost always preferred.
Multiply by 180/π
Write it as: (radian value × 180) / π. The π in the radian value and the π in the denominator cancel each other out.
Simplify — π cancels
For exact radian values (anything with π), the π cancels and you get a clean number × 180 / denominator. Multiply out and you have degrees.
Worked Examples
| Radians | × 180/π | π cancels, simplify | Result |
|---|---|---|---|
| π/6 | (π/6) × (180/π) | 180/6 | 30° |
| π/4 | (π/4) × (180/π) | 180/4 | 45° |
| 3π/2 | (3π/2) × (180/π) | 3 × 90 | 270° |
| 7π/6 | (7π/6) × (180/π) | 7 × 30 | 210° |
| 5π/3 | (5π/3) × (180/π) | 5 × 60 | 300° |
Full Reference Table — All 16 Unit Circle Angles
These are the 16 standard angles that appear on the unit circle and in every trig course. Bookmark this table — it’s the one you’ll come back to.
All 16 standard angles in both units. Blue = degrees, green = radians.
| Degrees | Radians | Quadrant | Decimal (rad) |
|---|---|---|---|
| 0° | 0 | Axis | 0.000 |
| 30° | π/6 | Q I | 0.524 |
| 45° | π/4 | Q I | 0.785 |
| 60° | π/3 | Q I | 1.047 |
| 90° | π/2 | Axis | 1.571 |
| 120° | 2π/3 | Q II | 2.094 |
| 135° | 3π/4 | Q II | 2.356 |
| 150° | 5π/6 | Q II | 2.618 |
| 180° | π | Axis | 3.142 |
| 210° | 7π/6 | Q III | 3.665 |
| 225° | 5π/4 | Q III | 3.927 |
| 240° | 4π/3 | Q III | 4.189 |
| 270° | 3π/2 | Axis | 4.712 |
| 300° | 5π/3 | Q IV | 5.236 |
| 315° | 7π/4 | Q IV | 5.498 |
| 330° | 11π/6 | Q IV | 5.760 |
| 360° | 2π | Axis | 6.283 |
Recognizing Radian Form Without Converting
Memorizing the conversion formulas is one skill. Recognizing what a radian value means at a glance is another — and it’s what actually gets tested. When you see 7π/6 on an exam, you should immediately know it’s 210°, in Quadrant III, without doing the arithmetic. Here’s how to build that pattern recognition.
The denominator tells you the family:
| Denominator | Degree family | Members |
|---|---|---|
| /6 | 30° family | π/6=30°, 5π/6=150°, 7π/6=210°, 11π/6=330° |
| /4 | 45° family | π/4=45°, 3π/4=135°, 5π/4=225°, 7π/4=315° |
| /3 | 60° family | π/3=60°, 2π/3=120°, 4π/3=240°, 5π/3=300° |
| /2 | 90° family | π/2=90°, 3π/2=270° (axis angles) |
The numerator tells you which member of the family: π/6 is the first 30°-family angle (30°), 5π/6 is the second (150°), 7π/6 is the third (210°), 11π/6 is the fourth (330°). As the numerator increases, you move counterclockwise around the circle.
Quadrant shortcut from the numerator: For the /6 family — numerator 1 = Q I, numerator 5 = Q II, numerator 7 = Q III, numerator 11 = Q IV. You can build similar shortcuts for each family once you spend a few minutes with the reference table above.
Negative Angles and Angles Beyond 2π
Standard conversion problems use angles between 0 and 2π. But exams — especially in pre-calculus and trigonometric equations — regularly throw negative angles and values larger than 2π. The conversion formulas work exactly the same way; the only adjustment is understanding what those angles represent geometrically.
Negative angles rotate clockwise rather than counterclockwise. −π/6 and 11π/6 land at exactly the same point on the unit circle (330°) — one traveled clockwise, one counterclockwise. To convert a negative radian to a positive equivalent, add 2π. To convert a negative degree to a positive equivalent, add 360°.
Examples with negative angles:
- −π/6 radians = −30° (clockwise from 0°, lands at 330°)
- −π/2 radians = −90° (clockwise from 0°, lands at 270°)
- −π/6 + 2π = 11π/6 (the positive coterminal equivalent)
- −45° × π/180 = −π/4 (conversion formula works identically)
Angles beyond 2π (or beyond 360°) simply wrap around the circle. 5π/2 is the same terminal position as π/2 — it’s just one full rotation plus π/2. To find the equivalent angle between 0 and 2π, subtract 2π until you’re in range. The conversion formulas still apply directly: 5π/2 × 180/π = 450°, which reduces to 450° − 360° = 90°.
💡 Coterminal Angles
Two angles are coterminal if they share the same terminal side — meaning they land at the same point on the unit circle. To find all coterminal angles, add or subtract multiples of 2π (radians) or 360° (degrees). This concept appears heavily in solving trig equations, where solutions repeat every full rotation.
Calculator Mode: Degrees vs Radians
This is not a minor detail. Getting the calculator mode wrong produces answers that are completely wrong — not slightly off, but wrong by a factor that makes no sense in context. It’s one of the most common reasons students lose full credit on trig problems.
How to check and change mode on common tools:
TI-84
- Press MODE
- Arrow down to the row with RADIAN / DEGREE
- Highlight DEGREE for degrees, RADIAN for radians
- Press ENTER to confirm
- Current mode shows in top of screen
Desmos
- Click the wrench icon (top right)
- Under Trigonometry, toggle between Degrees and Radians
- Mode shows in the settings panel
- Default is radians — check before assuming
iPhone / Android Calculator
- Open the scientific calculator (rotate to landscape)
- Look for Rad or Deg button
- Tap to toggle between modes
- Current mode is displayed on the button itself
⚠️ Quick Sanity Check
Type sin(30) into your calculator. If the answer is 0.5, you’re in degree mode. If the answer is −0.988, you’re in radian mode (because sin(30 radians) ≠ sin(30°)). Do this check before starting any trig problem.
The Calculus Connection
If you’re in pre-calculus or heading into calculus, there’s a concrete reason why radians matter beyond just converting numbers — and understanding it makes radians feel necessary rather than arbitrary.
The derivative of sin(x) is cos(x). But that rule only holds when x is measured in radians. In degree mode, the derivative of sin(x°) is (π/180)cos(x°) — a messier formula with an extra constant cluttering every calculation. Radians eliminate that constant because they’re defined in terms of the natural geometry of the circle, not an arbitrary 360-degree convention.
The same issue appears in limits. The fundamental limit that makes all of trig calculus work — lim(x→0) of sin(x)/x = 1 — is only true in radians. In degrees, that limit equals π/180. Calculus was built assuming radians, which is why every calculus course, every physics course, and every engineering application uses radians as the default.
💡 The Practical Takeaway
In pre-calc: your course will tell you which unit to use for each topic. Most unit circle and graphing problems use radians. Most right triangle and introductory problems use degrees. In calculus and beyond: assume radians unless told otherwise. Always.
Common Mistakes
❌ Inverting the formula
Multiplying by 180/π when you wanted π/180, or vice versa. The memory check: radians contain π, so the formula that produces radians must introduce π (× π/180). If your answer should be in radians and π disappeared from your answer, you used the wrong formula.
❌ Converting π to 3.14159 too early
Nearly all textbook and platform problems want exact radian answers — π/6, not 0.5236. Don’t substitute a decimal for π until the problem explicitly asks for a decimal approximation. Entering 0.785 instead of π/4 on Knewton Alta or Hawkes will be marked wrong.
❌ Forgetting to simplify the fraction
180π/180 should simplify to π, not be left as a fraction. 60π/180 should simplify to π/3. Always reduce. Platforms like ALEKS and MyMathLab often mark un-simplified answers incorrect even if the value is technically equivalent.
❌ Treating radians as if they have units
Radians are dimensionless — they’re a ratio (arc length divided by radius), not a unit like degrees. You don’t write “2π radians/second” the same way you write “360 degrees/second.” In most formal expressions and calculus contexts, the word “radians” is dropped entirely. This trips students up when they see bare numbers like π/3 and don’t recognize them as angle measurements.
❌ Wrong calculator mode
Covered above but worth repeating: sin(30°) = 0.5, sin(30 radians) = −0.988. These are completely different values. A wrong calculator mode turns a correct setup into a wrong answer with no obvious error to spot.
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Practice Problems
📝 Problem 1 — Degrees to Radians
Convert 225° to radians. Give your answer in exact form.
Click to reveal answer
225° × π/180 = 225π/180
GCD of 225 and 180 is 45: 225 ÷ 45 = 5, 180 ÷ 45 = 4
Answer: 5π/4 (Quadrant III, equivalent to 225°)
📝 Problem 2 — Radians to Degrees
Convert 11π/6 to degrees.
Click to reveal answer
(11π/6) × (180/π) = (11 × 180)/6
π cancels → 11 × 30 = 330°
Quick check: 11π/6 has denominator 6 → 30° family. Numerator 11 → fourth member of the family → 330°. ✓
📝 Problem 3 — Negative Angle
Convert −π/3 to degrees and find the positive coterminal equivalent in both degrees and radians.
Click to reveal answer
Convert: (−π/3) × (180/π) = −180/3 = −60°
Positive coterminal in degrees: −60° + 360° = 300°
Positive coterminal in radians: −π/3 + 2π = −π/3 + 6π/3 = 5π/3
Platform Tips
Knewton Alta almost always requires exact radian answers with π rather than decimal approximations. If your answer is π/4, type it as a fraction — not 0.785. Knewton uses an equation editor that accepts π as a symbol, so use it.
Hawkes Learning Certify mode will repeat radian conversion problems until you demonstrate mastery. The most common failure point is entering un-simplified fractions — Hawkes expects π/3, not 60π/180. Simplify before entering.
MyMathLab (Pearson) sometimes presents radian inputs in multiple-choice format showing both exact and decimal options. Choose the exact form (π/6, π/4, etc.) unless the question specifically says “to the nearest hundredth.”
ALEKS radian problems frequently include a fraction-entry interface — type the numerator, select the fraction bar, type the denominator, then add π. If you’re unsure how the input works, use the “help me solve this” option on a practice problem to see the expected format before it counts against you on a Knowledge Check.
Frequently Asked Questions
Why do we use radians instead of just degrees?
Degrees are a historical convention — someone divided a circle into 360 parts (likely based on the approximate number of days in a year). Radians are based on the actual geometry of circles: the ratio of arc length to radius. That geometric definition makes radians the natural choice for calculus, physics, and engineering because trig functions and their derivatives have simpler forms in radians. The derivative of sin(x) is cos(x) only in radians.
How do I remember which formula converts which way?
Radians contain π — so the formula that produces radians must introduce π. Multiply by π/180 to get radians. The formula that removes π (× 180/π) gives you degrees. If you blank, derive it: 360° = 2π radians → 1° = π/180 radians → multiply both sides by whatever degree value you have.
What does a radian look like as a decimal?
1 radian ≈ 57.3°. A full circle is 2π ≈ 6.28 radians. Common reference points: π/2 ≈ 1.57 radians (90°), π ≈ 3.14 radians (180°), 3π/2 ≈ 4.71 radians (270°). If you see a decimal between 0 and 6.28 used as an angle without a degree symbol, it’s almost certainly in radians.
Do I need to convert between radians and degrees in calculus?
Rarely — calculus works almost exclusively in radians. You’ll convert to degrees occasionally to check your intuition (is this angle in Q I or Q III?) but almost never in a calculation. The formulas for derivatives, integrals, and limits of trig functions all assume radians. If you enter a degree value where radians are expected, your entire calculation will be wrong.
How is radian conversion related to the unit circle?
The unit circle labels all 16 standard angles in both degrees and radians. Radian conversion is the skill that lets you move fluently between those two labels. Once you know that π/6 = 30° and that the /6 family corresponds to the 30° family, you can read unit circle coordinates in either notation without stopping to calculate. See our Unit Circle Explained guide for the full coordinate table.
What does it mean if a radian value has no π in it?
A radian value without π is a decimal approximation rather than an exact value. For example, 1.047 radians ≈ π/3 ≈ 60°. Decimal radians appear in applied problems (physics, engineering) and when using a calculator in radian mode. In pure trig courses, you’ll almost always work with exact values like π/4 or 3π/2 rather than decimals.
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Related Resources
In this cluster:
- Do My Trigonometry Homework
- Unit Circle Explained — all 16 angles with (cos θ, sin θ) coordinates in both units
- SOH-CAH-TOA Explained — right triangle trig using degrees
Where radians are used next:
- Graphing Sine and Cosine — period expressed in radians (2π)
- Solving Trig Equations — solutions given in radians
- Inverse Trig Functions — outputs in radians by convention