Reference Angle Calculator

Work out reference angle instantly with clear inputs, formula shown and shareable results.

About Reference Angle Calculator

The reference angle is the acute angle between the terminal arm and the x-axis. It is found by reducing the angle to 0°–360° and then subtracting from or to the nearest multiple of 180°, which lets you read trig values off the first quadrant.

Formula

By quadrant

Q1: θ, Q2: 180° − θ, Q3: θ − 180°, Q4: 360° − θ

Worked example

With the default values below, Reference Angle Calculator returns Reference angle of 30.0000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Angle: 210
Angle unitone of Degrees, Radians, Gradians: Degrees

Results

Reference angle: 30.0000 °
Quadrant: Quadrant 3
Angle reduced to 0°–360°: 210.0000 °
Reference angle in radians: 0.52359878

Step by step

1Values used
2By quadrant
3Reference angle= 30.0000 °
4Quadrant= Quadrant 3
5Angle reduced to 0°–360°= 210.0000 °
6Reference angle in radians= 0.52359878

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is the reference angle of 210°?

210° is in quadrant 3, so the reference angle is 210 − 180 = 30°.

How does it help?

sin 210° = −sin 30° = −0.5. The reference angle gives the magnitude and the quadrant gives the sign.