🔄 Converters
📐 📐 Angle Converter: Degrees, Radians, and Gradians
Learn how to convert angles between degrees, radians, and gradians. Covers the conversion formulas, why radians exist, the unit circle, and common angle reference values.
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Angles can be expressed in three main units: degrees (everyday use), radians (mathematics and physics), and gradians (surveying). Most people are comfortable with degrees, but radians are essential in calculus, trigonometry, and physics because they make formulas cleaner and more natural. Understanding why radians exist — not just how to convert to them — makes the unit genuinely intuitive.
The Three Angle Units
- Degrees (°): A full circle = 360°. The degree system dates to ancient Babylonian astronomy, which used base-60 arithmetic. There's no deep mathematical reason for 360 — it was chosen because 360 has many factors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12...), making it convenient for dividing circles.
- Radians (rad): A full circle = 2π radians ≈ 6.2832 rad. One radian is the angle subtended when the arc length equals the radius. Radians are the natural unit for angles in mathematics — they make calculus derivatives of trig functions clean (d/dx sin(x) = cos(x) only holds when x is in radians).
- Gradians (grad / gon): A full circle = 400 gradians. A right angle = 100 gradians. Developed during the French Revolution as part of the metric system attempt. Still used in surveying, geodesy, and some European engineering contexts.
Conversion Formulas
| Convert |
Formula |
| Degrees → Radians | rad = deg × π / 180 |
| Radians → Degrees | deg = rad × 180 / π |
| Degrees → Gradians | grad = deg × 10 / 9 |
| Gradians → Degrees | deg = grad × 9 / 10 |
| Radians → Gradians | grad = rad × 200 / π |
Common Angle Reference Values
| Degrees |
Radians |
Gradians |
Name |
| 0° | 0 | 0 | Zero angle |
| 30° | π/6 | 33.33 | — |
| 45° | π/4 | 50 | — |
| 60° | π/3 | 66.67 | — |
| 90° | π/2 | 100 | Right angle |
| 180° | π | 200 | Straight angle |
| 270° | 3π/2 | 300 | — |
| 360° | 2π | 400 | Full circle |
Why Radians Are Used in Mathematics
Radians arise naturally from the definition of arc length. For a circle of radius r, an arc subtending angle θ (in radians) has length s = rθ. This simple relationship makes radians the natural choice in calculus and physics:
- d/dx sin(x) = cos(x) — only true when x is in radians
- Small angle approximation: sin(θ) ≈ θ for small θ — only in radians
- Angular velocity ω (rad/s) × radius r = linear velocity v (m/s)
- The Taylor series for sin(x) = x − x³/6 + x⁵/120 − ... — only valid in radians
Calculator Mode Reminder
Scientific calculators can operate in DEG or RAD mode. A common error: computing sin(90) in RAD mode returns 0.894 instead of 1. Always check your calculator's mode before using trig functions. In programming, most math libraries (Python's math module, JavaScript's Math object) use radians by default.
❓ Frequently Asked Questions
How do I convert degrees to radians?▼
Multiply degrees by π/180 (approximately 0.01745). Examples: 90° × π/180 = π/2 ≈ 1.5708 rad; 180° = π rad; 360° = 2π rad. To remember: a full circle is 360° = 2π rad, so multiply degrees by 2π/360 = π/180.
How do I convert radians to degrees?▼
Multiply radians by 180/π (approximately 57.296). Examples: π rad × 180/π = 180°; π/2 rad = 90°; 1 rad ≈ 57.3°. One radian is the angle where arc length equals the radius — about 57.3° on a standard circle.
What is a gradian (grad)?▼
A gradian divides a full circle into 400 equal parts, making a right angle exactly 100 gradians. Developed during the French Revolution as part of the metric system. Still used in surveying, geodesy, and some European engineering. To convert: degrees × 10/9 = gradians (90° = 100 grad, 180° = 200 grad).
Why do mathematicians use radians instead of degrees?▼
Radians make calculus formulas clean and natural. The derivative of sin(x) is cos(x) only when x is in radians. The small-angle approximation sin(θ) ≈ θ only works in radians. Arc length formula s = rθ requires radians. Degrees work fine for everyday geometry but create messy constants (π/180) everywhere in higher mathematics.
Why does my calculator give the wrong trig result?▼
Your calculator is almost certainly in the wrong angle mode. sin(90) should equal 1 in degree mode but gives ≈0.894 in radian mode (since 90 radians is about 14.3 full circles). Check for DEG/RAD/GRAD indicator on your display. Most scientific calculators toggle modes via a DRG or MODE button.