Radians to Degrees: Formula, Conversion Table, and Worked Examples

To convert radians to degrees, multiply by 180 and divide by π. That is the whole formula. But if you have ever typed an angle into a calculator and gotten a completely wrong answer, or wondered why your programming language returns nonsense from a trig function, the reason almost always comes down to mixing up radians and degrees without realizing it.

This guide gives you the formula, a full reference table for the angles you will actually use, step-by-step worked examples, and a clear answer to why two systems exist in the first place.

Table of Contents

• The Radians to Degrees Formula
• Worked Examples
• Common Angles: Radians to Degrees Table
• Degrees to Radians: Reverse Conversion
• What Is a Radian?
• Why Radians Exist: Calculus, Physics, and Code
• When to Use Degrees vs Radians
• Common Calculator and Code Mistakes
• Frequently Asked Questions

The Radians to Degrees Formula

The conversion formula is:

Degrees = Radians × (180 / π)

Since π ≈ 3.14159265, the decimal conversion factor is:

Degrees = Radians × 57.2958

Both forms give the same result. The first is more exact because it keeps π symbolic. The second is faster for mental estimation.

The formula comes from one fact: a full circle is both 360° and 2π radians. Divide both sides by 2π and you get 1 radian = 180/π degrees.

Worked Examples

Example 1: Convert π/6 radians to degrees

Degrees = (π/6) × (180/π). The π cancels: Degrees = 180/6 = 30°

Example 2: Convert π/4 radians to degrees

Degrees = (π/4) × (180/π) = 180/4 = 45°

Example 3: Convert π/3 radians to degrees

Degrees = (π/3) × (180/π) = 180/3 = 60°

Example 4: Convert 2.5 radians to degrees

Degrees = 2.5 × (180/π) = 2.5 × 57.2958 = 143.24°

Example 5: Convert 5 radians to degrees

Degrees = 5 × (180/π) = 5 × 57.2958 = 286.48°

Example 6: Convert a negative angle — −π/2 radians to degrees

Degrees = (−π/2) × (180/π) = −180/2 = −90°

Negative radians rotate clockwise. −90° occupies the same position as 270°, measured in the opposite direction.

Common Angles: Radians to Degrees Table

These are the angles you meet repeatedly in trigonometry, calculus, and physics. Knowing the first-quadrant set (30°, 45°, 60°, 90°) by memory covers the majority of problems.

Radians (exact)	Radians (decimal)	Degrees	Description
0	0	0°	Zero angle
π/6	0.5236	30°	One-twelfth of a circle
π/4	0.7854	45°	One-eighth of a circle
π/3	1.0472	60°	One-sixth of a circle
π/2	1.5708	90°	Right angle
2π/3	2.0944	120°	One-third of a circle
3π/4	2.3562	135°	Three-eighths of a circle
5π/6	2.6180	150°	Five-twelfths of a circle
π	3.1416	180°	Straight line / half circle
7π/6	3.6652	210°	Third quadrant start
5π/4	3.9270	225°	Southwest diagonal
4π/3	4.1888	240°	Two-thirds of a circle
3π/2	4.7124	270°	Three-quarter turn
5π/3	5.2360	300°	Five-sixths of a circle
7π/4	5.4978	315°	Seven-eighths of a circle
11π/6	5.7596	330°	Eleven-twelfths of a circle
2π	6.2832	360°	Full circle

Memory shortcut: for the first-quadrant angles, the radian denominators are 6, 4, 3, and 2. These match the denominators of the fractions of a right angle. Once you know those four, the rest of the unit circle follows by quadrant symmetry.

Degrees to Radians: Reverse Conversion

To go the other direction:

Radians = Degrees × (π / 180)

Or using the decimal factor:

Radians = Degrees × 0.017453

Three quick examples:

• 45° = 45 × (π/180) = π/4 ≈ 0.7854 rad
• 90° = 90 × (π/180) = π/2 ≈ 1.5708 rad
• 270° = 270 × (π/180) = 3π/2 ≈ 4.7124 rad

What Is a Radian?

A radian is the angle at the center of a circle where the arc length equals the radius. Draw a circle with radius r. Mark an arc along the edge that is also length r. The angle at the center subtended by that arc is exactly 1 radian.

That definition makes the radian a ratio of two lengths — arc length divided by radius — which means it carries no units. No conversion factor cancels out when you use radians in a formula. Degrees, by contrast, are an arbitrary division of the circle into 360 parts, a number chosen by Babylonian astronomers roughly 4,000 years ago because 360 has 24 divisors and approximates the number of days in a year.

Because a full circle has circumference 2πr, and the radius is r, the number of radii that fit around the circumference is exactly 2π. So a full circle contains 2π radians, which equals 360°. Every conversion follows from that:

• 2π rad = 360°
• π rad = 180°
• 1 rad = 180/π ≈ 57.2958°

Why Radians Exist: Calculus, Physics, and Code

Degrees feel natural for everyday geometry. A right angle is 90°, a straight line is 180°, a full turn is 360°. So why do mathematicians, physicists, and programmers default to radians?

The answer is calculus.

The derivative of sin(x) is cos(x) — but only when x is in radians. In degrees, the derivative of sin(x) becomes (π/180) × cos(x). That correction factor propagates through every derivative and integral involving a trig function. Working in degrees means carrying π/180 through every calculation. Radians eliminate it entirely.

The same pattern holds across physics and engineering:

• Angular velocity (ω) is measured in radians per second. The relationship between linear velocity and angular velocity is v = rω. This formula requires ω in rad/s. In degrees per second, the answer is wrong by a factor of π/180.
• Wave equations use radian angular frequency. A sine wave written as sin(ωt) assumes ω is in radians per second.
• Arc length is s = rθ, where θ must be in radians. In degrees, the formula gains a factor: s = rθ(π/180).
• Small-angle approximation in physics: sin(θ) ≈ θ for small θ. This approximation holds only in radians. A pendulum with a 5° swing is 0.0873 radians, and sin(0.0873) ≈ 0.0872 — close enough for most calculations. In degrees, sin(5) and 5 are nowhere near each other.

Programming languages inherit the mathematical convention. Python’s math.sin(), JavaScript’s Math.sin(), C’s sin(), and every standard library trig function takes radians. The underlying numerical methods (Taylor series, CORDIC algorithms) all assume radian input.

When to Use Degrees vs Radians

Both units are correct. The question is which one fits the task.

Use degrees for:

• Everyday geometry — measuring angles in construction, carpentry, and architecture
• Navigation — compass headings, aircraft bearing, latitude and longitude all use degrees
• Communication — saying “turn 90° to the right” is clearer than “turn π/2 radians”
• Surveying and land measurement — theodolites read in degrees, minutes, and seconds

Use radians for:

• Calculus — derivatives, integrals, and limits of trig functions
• Physics — angular velocity, angular acceleration, wave equations, pendulum motion
• Computer graphics — rotation matrices, 3D transformations, WebGL, OpenGL
• Signal processing — Fourier transforms and frequency analysis use radian frequency
• Robotics — joint angles and kinematics throughout
• Programming — any call to a standard library trig function (sin, cos, tan, atan2)

Quick rule: if you are working with formulas involving derivatives, arc length, or angular velocity, use radians. If you are describing an angle to a person or reading it from a protractor, degrees are fine.

Common Calculator and Code Mistakes

Most wrong trig answers come from one of four places.

Calculator in the wrong mode. Scientific calculators have a DEG/RAD/GRAD switch. If your calculator is in DEG mode and you type sin(π/2), you get sin(1.5708°), which is about 0.0274, not 1. Check the mode before every trig calculation.

Excel SIN() with degrees. Excel’s built-in trig functions take radians. =SIN(30) computes the sine of 30 radians, which is approximately −0.988. To get sin(30°), write =SIN(RADIANS(30)) or =SIN(30*PI()/180).

Mixing units mid-calculation. A common error in physics problems: computing sin(θ) where θ is in degrees, then using the result in a formula that assumes radians. Convert all angles to one unit at the start of a problem and stay consistent throughout.

Negative angle direction. Negative radians represent clockwise rotation. −π/4 radians sits at the same position as 315° measured counterclockwise. In navigation and some engineering contexts, angles always run 0° to 360°, so a result of −30° may need to be reported as 330°.

Frequently Asked Questions

How do you convert radians to degrees?

Multiply the radian value by 180 and divide by π (approximately 3.14159). The formula is: degrees = radians × (180 / π). For example, π/2 radians equals 90°.

How many degrees is 1 radian?

1 radian equals approximately 57.2958 degrees. This comes from dividing 180 by π: 180 / 3.14159 = 57.2958.

What is π radians in degrees?

π radians equals exactly 180 degrees. This is the most useful conversion anchor: half a circle is π radians and 180°.

How do you convert degrees to radians?

Multiply the degree value by π and divide by 180. The formula is: radians = degrees × (π / 180). For example, 90° equals π/2 radians, approximately 1.5708 rad.

Why do calculators and programming languages use radians instead of degrees?

Radians are the natural unit for trigonometric functions in calculus. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, a correction factor of π/180 appears in every derivative and integral. Programming languages like Python, JavaScript, and C inherit this convention from mathematics.

What is the difference between radians and degrees?

Degrees divide a full circle into 360 equal parts, a number chosen historically because it has many divisors. Radians measure an angle by the arc it subtends on a unit circle: one radian is the angle where the arc length equals the radius. A full circle is 360° or 2π radians (approximately 6.283).

Should I use radians or degrees in Excel?

Excel’s SIN(), COS(), and TAN() functions expect radians. =SIN(30) gives the sine of 30 radians, not 30 degrees. To use degrees, write =SIN(RADIANS(30)) or =SIN(30*PI()/180).