One degree equals π/180 radians, so to convert an angle from degrees to radians you multiply the number of degrees by π/180 (equivalently, multiply by 0.017453292519943295). For example, 90° becomes 90 × π/180 = π/2 ≈ 1.5707963267948966 radians. The conversion works because a full revolution is 360°, the same as 2π radians, so the ratio 2π/360 simplifies to π/180. A pocket calculator or a dedicated tool can do this multiplication for you, but the formula itself is worth knowing because trigonometry functions in spreadsheets, programming languages, and physics formulas are almost always written to expect radian input.
The two systems measure the same thing — the amount of rotation between two rays — but they slice a circle differently. Degrees carve the circle into 360 equal pieces, a number inherited from ancient Babylonian astronomy. Radians are defined as the arc length of a unit circle subtended by the angle, which is why one full revolution equals exactly 2π. That makes radians the "natural" unit for calculus and physics: derivatives of sine and cosine only work cleanly when the argument is in radians, and formulas like angular velocity ω = v/r or arc length s = rθ are written assuming θ is in radians. Knowing how to move back and forth between the two is a basic skill for anyone working with trigonometric data.
The Degrees-to-Radians Formula
The conversion rests on one identity:
180 degrees = π radians
Dividing both sides by 180 gives the unit conversion factor:
1 degree = π/180 radians ≈ 0.017453292519943295 radians
To convert any angle θ in degrees to radians, compute:
θ (radians) = θ (degrees) × π/180
Multiplying by π/180 is the same as dividing by 57.29577951308232, which is often useful when you want to think in terms of "how many radians fit in a degree." For common angles you will quickly memorize a few shortcuts:
- 30° = π/6 ≈ 0.5236 rad
- 45° = π/4 ≈ 0.7854 rad
- 60° = π/5 ≈ 1.0472 rad
- 90° = π/2 ≈ 1.5708 rad
- 180° = π ≈ 3.1416 rad
- 360° = 2π ≈ 6.2832 rad
These six reference angles cover most textbook problems and they are also a handy sanity check whenever you convert a value by hand.
Convert a Degree Value to Radians Step by Step
For a quick worked example, take 75° and walk through the formula in detail.
Step 1 — Identify the value. You are converting 75 degrees to radians.
Step 2 — Write the formula. θ_rad = θ_deg × π/180.
Step 3 — Substitute. θ_rad = 75 × π/180 = 75π/180.
Step 4 — Simplify the fraction. 75π/180 reduces to 5π/12 (dividing both 75 and 180 by their greatest common divisor, 15).
Step 5 — Evaluate numerically. 5 × 3.141592653589793 ÷ 12 = 15.707963267948966 ÷ 12 ≈ 1.308996938995747 radians.
So 75° equals exactly 5π/12 radians, or about 1.309 rad. The answer in radians is always a smaller-looking decimal than the original degree value because π/180 is a small factor. A quick mental check: anything between 0° and 180° should land between 0 and π, and 1.309 sits comfortably in that range, so the result is plausible.
Convert Degrees to Radians With the Angle Converter
For everyday conversions — especially when the angle has many decimal places, is negative, or you also want the value in gradians, turns, arcminutes, or arcseconds at the same time — the Angle Converter tool does the work in a single screen. Here is the exact workflow:
- Open the Angle Converter page and locate the Value field at the top.
- Type the angle you want to convert. Decimals (for example 47.25) and negative numbers (for example -120) are both accepted.
- Under From, choose the source unit. For a degree-to-radian conversion, select Degrees (°).
- Under To, choose the target unit. For radians, select Radians (rad).
- Read the converted result, which updates as soon as you finish typing. If you need the inverse, click Swap to flip the From and To units without retyping anything.
- Scroll to the comparison table below the result to see the same angle expressed in every supported unit — degrees, radians, gradians, turns, arcminutes, and arcseconds — at once.
This workflow covers the typical "I have a degree value and I need radians" case, but it also handles the reverse direction, mixed-unit comparisons, and angle inputs that come from geographic coordinates, mechanical drawings, or surveying data.
When You Need Other Angle Units Too
Degrees and radians are the two units most people meet in school, but several other systems show up in specific fields:
| Unit | Symbol | One full revolution | Common context |
|---|---|---|---|
| Degree | ° | 360° | Navigation, geography, everyday use |
| Radian | rad | 2π rad | Math, physics, engineering formulas |
| Gradian (gon) | gon | 400 gon | Surveying and some European engineering standards |
| Turn (revolution) | tr | 1 tr | Robotics, mechanical rotations, CSS transforms |
| Arcminute | ' | 21,600′ | Astronomy, GPS coordinates, surveying |
| Arcsecond | " | 1,296,000″ | Astronomy, fine angular measurement |
The Angle Converter handles every row in this table, which is especially handy when a problem hands you an angle in gradians or turns and you need it in radians, or vice versa. The unit definitions themselves are standardized by the International Organization for Standardization, so the values shown by the tool match what any well-calibrated scientific calculator would produce.
Common Situations Where the Conversion Comes Up
Several real-world tasks rely on a clean degree-to-radian conversion, and knowing the formula — or having a tool ready — saves time in each.
- Spreadsheet trigonometry. Excel's SIN, COS, TAN, ASIN, and ACOS functions all expect radians. If your data is in degrees, you have to multiply by π/180 first (or wrap it in the RADIANS function) before feeding it in. This is the most common reason non-mathematicians search for the conversion at all.
- Programming and scripting. JavaScript's Math.sin and Math.cos, Python's math module trig functions, and most numerical libraries work in radians. Converting degrees to radians is therefore the first step of nearly every geometry or graphics routine.
- Physics formulas. Angular velocity, period, frequency, centripetal acceleration, and arc length all use θ in radians. A 30° pendulum swing is not "30" in those formulas — it is π/6.
- Engineering and surveying. Civil engineers converting between azimuths in degrees and computations in radians for coordinate transformations, or surveyors working with gradians and needing radians for downstream math.
- Geography and astronomy. Latitude and longitude are reported in degrees, minutes, and seconds, but calculations on the surface of a sphere (great-circle distance, bearing) are done in radians.
For any of these situations, the Angle Converter produces the precise radian value without requiring you to keep π memorized or to worry about whether your calculator is set to the right mode.
Tips for Avoiding Conversion Mistakes
Even with a simple factor like π/180, errors creep in. A few habits keep things accurate:
- Check your calculator mode. Many scientific calculators have a DEG/RAD/GRAD toggle. If you get an answer that is wildly off (for example, sin(30°) returning 0.988 instead of 0.5), the mode is almost always the culprit.
- Sanity-check the size. A degree value between 0 and 360 should give a radian value between 0 and about 6.283. A result outside that range signals a wrong multiplication or a unit mix-up.
- Watch for decimal places. One degree is about 0.0174533 rad, so a 10° angle is roughly 0.175 rad — not 0.0175 rad. Place-value slips are the most common source of off-by-100 errors.
- Remember negative angles. Negative degrees convert just as cleanly: -45° becomes -π/4 ≈ -0.7854 rad. The sign carries straight through the multiplication.
- Use the tool for non-standard units. If a problem involves gradians, turns, or arcminutes, skip the manual math and use the Angle Converter to get every unit on screen at once.
For related conversions that involve numbers rather than angles, the Scientific Calculator evaluates trig, logs, and roots directly in radians, which pairs naturally with the Angle Converter once you have the value in the right unit. And if your work also calls for area, length, or volume calculations that depend on those angles, the Circle Area Calculator and Circumference Calculator handle the downstream geometry.
Frequently Needed Reference Values
Below are some commonly used degree-to-radian conversions that are worth keeping within reach. They are derived directly from the formula θ_rad = θ_deg × π/180, so you can verify any of them by hand in a few seconds.
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 1° | π/180 | 0.01745 |
| 15° | π/12 | 0.26180 |
| 30° | π/6 | 0.52360 |
| 45° | π/4 | 0.78540 |
| 60° | π/3 | 1.04720 |
| 90° | π/2 | 1.57080 |
| 120° | 2π/3 | 2.09440 |
| 180° | π | 3.14159 |
| 270° | 3π/2 | 4.71239 |
| 360° | 2π | 6.28319 |
For values outside this list, or for angles that include decimal degrees, the Angle Converter returns the precise radian result without forcing you to do the π/180 multiplication in your head.
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