About

In trigonometry and calculus, accuracy depends on the unit of angular measurement. While degrees are intuitive for navigation and basic geometry, radians are the standard unit for higher mathematics because they simplify derivatives and integrals of trigonometric functions. A radian is defined by the radius of a circle wrapped along its circumference.

Students often face the challenge of converting decimal degrees into exact radians expressed in terms of π. Using a standard calculator results in a messy irrational number like 1.5707..., which is often unacceptable for academic work that requires the precise symbolic form π2. This tool automates the simplification process, finding the greatest common divisor between the input angle and 180 to provide the clean, fractional output required for homework and exams.

Formulas

The relationship between degrees and radians is derived from the circumference of a circle (C = 2πr). One complete revolution equals 360 degrees or 2π radians. Therefore, the conversion factor is defined as:

Rad = Deg × π180°

To obtain a symbolic result, we treat the input degree as the numerator of a fraction over 180 and simplify using the Greatest Common Divisor (GCD):

Simplified Fraction = Deg ÷ GCD(Deg, 180)180 ÷ GCD(Deg, 180) π

Reference Data

Degrees (°)	Radians (Symbolic)	Radians (Decimal)	Quadrant
0°	0	0.0000	Axis
30°	π6	0.5236	I
45°	π4	0.7854	I
60°	π3	1.0472	I
90°	π2	1.5708	Axis
120°	2π3	2.0944	II
135°	3π4	2.3562	II
150°	5π6	2.6180	II
180°	π	3.1416	Axis
210°	7π6	3.6652	III
225°	5π4	3.9270	III
270°	3π2	4.7124	Axis
315°	7π4	5.4978	IV
330°	11π6	5.7596	IV
360°	2π	6.2832	Axis

Frequently Asked Questions

Symbolic output retains exact precision. Decimal approximations (like 3.14...) introduce rounding errors that compound in complex derivatives or integrals. Keeping Pi as a symbol ensures the logic remains rigorous.

The formula applies identically to negative values. A negative angle simply indicates rotation in the clockwise direction. For example, -90 degrees converts to -Pi/2 radians.

Yes. Angles greater than 360 degrees (coterminal angles) are converted normally. For instance, 720 degrees converts to 4Pi radians.

A degree is 1/360th of a circle's rotation, an arbitrary historical convention. A radian is the angle subtended when the arc length equals the radius, a physically intrinsic property of circle geometry.