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Reference Angle Calculator

Step 1 / 3 Reference Angle Calculator

Any angle, positive or negative, even beyond one full turn. In radians you can write π as pi, for example 5pi/6.

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The reference angle is the acute angle (or 90 degrees when the angle lies on the y-axis) that the terminal side of an angle makes with the x-axis. Type any angle, positive or negative, in degrees or radians, even one beyond a full turn such as -750 or 5π/6. The calculator brings it into the 0 to 360 degree range, names the quadrant, gives the reference angle in degrees and radians, lists the nearest coterminal angles and writes out the steps.

How to find a reference angle

  1. 1

    Enter the angle

    Type a number such as 135, -750 or 210.5. For radians you can type a plain number (2.5) or a multiple of π written with pi or π (5pi/6, -3*pi/4, 2π).

  2. 2

    Choose degrees or radians

    Pick the unit the angle is written in. A radian angle is converted to degrees first, and that conversion is shown in the steps.

  3. 3

    Read the reference angle

    The result page shows the reference angle in degrees and radians, the angle between 0 and 360 degrees, its quadrant, the coterminal angles and the working.

What the reference angle is

Draw an angle in standard position, starting on the positive x-axis. The reference angle is the smallest angle between its terminal side and the x-axis. It is never negative and never more than 90 degrees, so it lets you read the sine, cosine and tangent of any angle from the values of an angle in the first quadrant. Only the signs change from quadrant to quadrant.

The formulas

First bring the angle into the range 0 to 360 degrees by adding or subtracting full turns of 360 degrees (2π radians). Call the result θ. Then:

Where θ lies Reference angle
Quadrant I (0° to 90°) θ
Quadrant II (90° to 180°) 180° − θ
Quadrant III (180° to 270°) θ − 180°
Quadrant IV (270° to 360°) 360° − θ

In radians the same rules read θ, π − θ, θ − π and 2π − θ.

Worked examples

Angle Normalized Quadrant Reference angle
135° 135° II 45°
225° 225° III 45°
300° 300° IV 60°
-750° 330° IV 30°
400° 40° I 40°
7π/6 210° III 30° (π/6)
11π/6 330° IV 30° (π/6)

Angles on an axis

An angle of 0°, 90°, 180° or 270° lies on an axis, so it has no quadrant. This calculator reports the axis instead, and gives a reference angle of 0° for the x-axis (0° and 180°) and 90° for the y-axis (90° and 270°).

Coterminal angles

Two angles are coterminal when they end on the same terminal side, so they differ by a whole number of turns. The result page lists the angle in the 0 to 360 degree range together with the angle one turn below and one turn above it. Any other coterminal angle is one of these plus or minus further multiples of 360° (2π).

Common mistakes

  • Subtracting the angle from 180° for every quadrant. That only works in quadrant II.
  • Forgetting to remove full turns first. 400° ends where 40° ends, so its reference angle is 40°.
  • Treating a negative angle as already in the first quadrant. -30° ends in quadrant IV, so its reference angle is 30°.

Frequently Asked Questions

135° lies in quadrant II, so the reference angle is 180° − 135° = 45°. In radians that is π/4.

No. The reference angle is measured from the x-axis to the terminal side and lies between 0° and 90°, whatever the sign or size of the original angle. -750° and 330° both have a reference angle of 30°.

Choose Radians and type a number such as 2.5, or a multiple of π using pi or the π sign: pi/6, 5pi/6, -3*pi/4, 2π. Multiples of π are converted to degrees exactly.

An angle of 90° points along the y-axis, which is 90° away from the x-axis, so that is its reference angle. 0° and 180° lie on the x-axis and have a reference angle of 0°.

The angle is sent to our server to work out the result, and it is added to the page link as you move between steps so a result can be bookmarked or shared. It is not saved to any account or profile.

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