Degrees to Radians

How to Convert Degrees to Radians?

One degree equals approximately 0.01745 radians. To convert degrees to radians, multiply the degree value by π/180 (approximately 0.01745). This is the most fundamental angle conversion in mathematics, physics, and engineering. Every trigonometry course, every physics problem involving circular motion, and every computer graphics calculation requires radians. While degrees are the intuitive human unit for angles (a full circle is 360 degrees), radians are the natural mathematical unit (a full circle is 2π radians). Scientific calculators, programming languages, and mathematical formulas universally expect angles in radians. If you put a degree value into the sine function of most programming languages without converting first, you will get a completely wrong answer. Understanding this conversion is non-negotiable for anyone in STEM fields. It also matters in classroom work because exact answers are often written as fractions of π rather than decimals. Converting 150° to 5π/6 instead of 2.618 helps you recognize unit-circle values, compare equivalent angles, and avoid rounding error in later steps. It also explains why formulas for angular velocity, phase angle, and unit-circle identities are almost always stated in radians rather than degrees.

How to Convert Degree to Radian

1. Start with your angle in degrees.
2. Multiply the degree value by π/180 to get radians.
3. The result is your angle in radians.
4. For common angles, memorize the key values: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.
5. For a quick decimal estimate, multiply degrees by 0.01745. For example, 45° x 0.01745 = 0.7854 radians.

Real-World Examples

History of Degree and Radian

The degree has ancient Babylonian origins — the Babylonians used a base-60 (sexagesimal) number system and divided the circle into 360 parts, likely because 360 is close to the number of days in a year and is highly divisible. The radian was developed much later by mathematicians who needed a unit that simplified calculus and trigonometric identities. The term "radian" was first used by James Thomson in 1873, though the concept existed earlier. The radian became the SI standard for angle measurement because it makes derivatives and integrals of trigonometric functions clean: d/dx[sin(x)] = cos(x) only works when x is in radians. If x were in degrees, the derivative would include an extra factor of π/180.

Common Mistakes to Avoid

• Forgetting to convert degrees to radians before using trigonometric functions in programming. In Python, JavaScript, C++, and most languages, math.sin(), math.cos(), and math.tan() expect radians. Passing 90 instead of π/2 gives sin(90 radians) ≈ 0.894, not the expected 1.
• Confusing the conversion direction: multiplying by 180/π instead of π/180 converts radians to degrees (the opposite). If your result is larger than the degree value, you went the wrong way.
• Treating π as exactly 3.14. Using 3.14 instead of a more precise value introduces error. For 180°: 180 x 3.14/180 = 3.14, but the correct value is 3.14159... Use your language's built-in π constant for precision.
• Converting to a rounded decimal when the context expects an exact radian value. In trigonometry homework and symbolic math, 60° is usually better written as π/3 than 1.0472 because the exact form preserves clean identities and later simplifications.

Frequently Asked Questions