Quick answer

180° = π rad. Quadrant boundaries: 90° = π/2, 180° = π, 270° = 3π/2.

Formula

  • Degrees: α = 180° − θ (QII example)
  • Radians: α = π − θ (QII example)

Introduction

The Reference Angle Calculator lets you stay in deg, rad, or π rad without manual conversion, which is useful when homework switches units problem to problem.

Whether you use degrees or radians, the steps are normalize, label quadrant, subtract once. Only the benchmark numbers change.

Keep the four formulas visible in our reference angle formula guide while you practice conversions here.

π rad mode on the home tool treats your entry as a multiple of π, which matches how textbooks write 3π/4 instead of decimal radians.

Choose one unit per problem

High school geometry and trigonometry often use degrees. Calculus, physics, and the unit circle in precalculus often use radians. Pick the unit stated in the prompt and keep subtractions consistent through the whole line of work.

To convert, multiply by π/180° or 180°/π. One full turn is 360° or 2π, and half a turn is 180° or π.

Mixed-unit mistakes look like α = π − 135° without converting 135° to radians first. Write either all degrees or all radians for each problem.

Special angles such as 45° and π/4 are the same direction. Reference angles respect that equality when you stay in one unit system.

Worked conversions

  • 240° in QIII → α = 240° − 180° = 60°
  • 7π/6 in QII → α = π − 7π/6 = π/6
  • 135° in QII → α = 180° − 135° = 45° = π/4

Enter 7π/6 as 1.167 with π rad selected on the home calculator to verify the radian example (7/6 ≈ 1.167).

The degree and radian versions of the same direction should produce α values that convert to each other: 60° = π/3, 45° = π/4, 30° = π/6.

For a full set of quadrant samples in both units, work through reference angle examples and convert each α to the other unit as extra practice.

Unit conversion workflow

  1. Convert only if needed. If the problem is in degrees, stay in degrees. If it is in radians, stay in radians unless the prompt explicitly asks for both.
  2. Normalize in that unit. Use 360° or 2π as one full turn. For negative angles, add one turn in the same unit.
  3. Apply the quadrant rule. Use 180° or π consistently in the subtraction. QIV uses 360° − θ or 2π − θ, never a mix.
  4. Check with π rad mode. When the angle is written as 5π/6, enter 0.833 with π rad selected instead of converting to 2.618 decimal radians by hand.

Example: 3π/4

3π/4 is in QII because it lies between π/2 and π. In radians, α = π − 3π/4 = π/4.

In degrees, 3π/4 equals 135°, still in QII, and α = 180° − 135° = 45°. Since π/4 radians equals 45°, both unit paths agree.

Enter 0.75 with π rad selected on the calculator to represent 3π/4 and confirm QII with α = π/4 in the output.