Angle Converter: Degrees, Radians, and Gradians Explained
Angles can be measured in degrees, radians, or gradians. Most people only encounter degrees in daily life, but radians are the standard in calculus and physics, and gradians appear in surveying and some European traditions.
Degrees
The familiar system: a full circle is 360°. Right angle: 90°. Straight line: 180°.
Why 360? Historically attributed to the approximately 360 days in a year and the divisibility of 360 by many small integers (1, 2, 3, 4, 5, 6, 8, 9, 10, 12...).
Radians
In radians, a full circle is 2π ≈ 6.2832 radians. A right angle is π/2 ≈ 1.5708 radians.
Why radians? Because they make the math of circular motion, trigonometry, and calculus cleaner. The derivative of sin(x) is cos(x) — but only when x is in radians. In degrees, it's (π/180)cos(x). Radians eliminate that constant.
Key conversions:
- 0° = 0 rad
- 90° = π/2 rad ≈ 1.5708
- 180° = π rad ≈ 3.1416
- 270° = 3π/2 rad ≈ 4.7124
- 360° = 2π rad ≈ 6.2832
Formula: radians = degrees × (π/180)
Gradians (Gon)
A full circle is 400 gradians. A right angle is exactly 100 gradians. Used in surveying, civil engineering, and military applications in some countries because the decimal relationship simplifies certain calculations.
The advantage: 45° = 50 gradians, 90° = 100 gradians — cleaner numbers for slope calculations.
When You'll Actually Use Radians
If you're writing trigonometric functions in code (Python, JavaScript, etc.), the math functions take radians, not degrees. math.sin(90) gives you the wrong answer; math.sin(math.pi/2) gives you 1.0.
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Frequently Asked Questions
Why does my calculator give a different answer when I use degrees vs radians?
Most programming languages and scientific calculators require radians as input, not degrees. For example, math.sin(90) in Python returns 0.89, not 1.0, because 90 is interpreted as 90 radians, not 90 degrees. To get the correct answer, convert first: math.sin(π/2) or math.sin(90 × π/180) both equal 1.0.
How do I convert 45 degrees to radians?
Use the formula: radians = degrees × (π/180). For 45°: 45 × (π/180) = π/4 ≈ 0.7854 radians. This is useful when you need to input angles into trigonometric functions in code or scientific applications.
What's the difference between radians and gradians?
Radians divide a full circle into 2π (≈6.2832) units, while gradians divide it into 400 units. Gradians are rarely used except in surveying and some European engineering fields—they offer the advantage that 90° = exactly 100 gradians, which simplifies slope calculations. For most scientific and programming work, radians are the standard.
Why is 360 degrees a full circle instead of some other number?
The 360-degree system was historically based on approximately 360 days in a year, but it stuck around because 360 is highly divisible by many small numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, etc.), making it convenient for dividing circles into equal parts. This is purely traditional—radians and gradians are mathematically cleaner for most modern applications.
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