What is the reference angle?

Type an angle in degrees, in radians, or as a multiple of π. The reference angle calculator gives the acute angle between its terminal side and the x-axis, in degrees and as a multiple of π, with the coterminal angle from 0° to 360°, the quadrant, and a picture of the point on the unit circle.

Reference angle (°): 45

The reference angle is 45°.

Reference angle (rad): 0.7853981634
In terms of π: π/4
Coterminal angle (°): 225
Quadrant: Quadrant III
Point x (cos θ): -0.7071
Point y (sin θ): -0.7071
Reference point x: 0.7071
Reference point y: 0.7071
Working: 225° is coterminal with 225°; Quadrant III; reference angle = angle − 180° = 45°

Reference angle (°): 45. The reference angle is 45°.

Where does the angle end on the unit circle?

How it is worked out

How to calculate

Finds the reference angle of any angle in degrees, radians or a multiple of π: the acute angle to the x-axis, with the coterminal angle, the quadrant, and the point on the unit circle.

Example with the default inputs (Angle in Degrees, Angle 225): The reference angle is 45°.

Method: Find the coterminal angle c from 0° to under 360°; the reference angle is c in quadrant I, 180° − c in II, c − 180° in III and 360° − c in IV (the same in radians with π and 2π).

An angle on an axis has reference angle 0° (the x-axis) or 90° (the y-axis).
Degrees and multiples of π are worked exactly; radians use π to double precision.
Angles from −10⁹ to 10⁹ in degrees or radians, or −10¹² π to 10¹² π.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

Angle in Degrees, Angle 225 gives Reference angle (°) 45, In terms of π π/4, Coterminal angle (°) 225, Quadrant Quadrant III, Reference angle (rad) 0.785398.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02)
Angle in × π, Angle ÷ π 5/3 gives In terms of π π/3, Reference angle (°) 60, Coterminal angle (°) 300, Quadrant Quadrant IV.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02)
Angle in × π, Angle ÷ π -1/6 gives In terms of π π/6, Coterminal angle (°) 330, Quadrant Quadrant IV.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02)
Angle in Degrees, Angle 870 gives Coterminal angle (°) 150, Reference angle (°) 30, Quadrant Quadrant II, In terms of π π/6.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02)
Angle in Radians, Angle 4 gives Reference angle (rad) 0.858407, Quadrant Quadrant III, Reference angle (°) 49.183118.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02)
Angle in Degrees, Angle -90 gives Coterminal angle (°) 270, Reference angle (°) 90, Quadrant On the negative y-axis, In terms of π π/2.Source: OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (the reference angle is the acute angle between the terminal side and the horizontal axis: t in quadrant I, 180° − t in II, t − 180° in III, 360° − t in IV; 225° → 45°, 5π/3 → π/3, −π/6 → π/6), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-02)

How it works

For an angle θ, first find the coterminal angle c, the angle from 0° to under 360° (0 to under 2π) that ends in the same place: c = θ mod 360° (θ mod 2π for radians), always taken as 0 or more. Then the reference angle is:

Where c lies	Reference angle (degrees)	Reference angle (radians)
0° ≤ c ≤ 90° (quadrant I)	c	c
90° < c ≤ 180° (quadrant II)	180° − c	π − c
180° < c ≤ 270° (quadrant III)	c − 180°	c − π
270° < c < 360° (quadrant IV)	360° − c	2π − c

the reference angle in radians = degrees × π ÷ 180, and in terms of π (when the reference angle in degrees is a whole number, or the angle was typed as a multiple of π) as a fraction in lowest terms, such as π/4 or 2π/9
the unit-circle point is (cos c, sin c); the reference point is (cos ref, sin ref) in quadrant I
quadrant: "Quadrant I" to "Quadrant IV" by the ranges above with strict ends, or "On the positive x-axis" (c = 0°), "On the positive y-axis" (90°), "On the negative x-axis" (180°), "On the negative y-axis" (270°)

Rules

Degrees: c = ((θ mod 360) + 360) mod 360. A multiple of π typed as a fraction n/d is worked exactly in whole numbers (c = n mod 2d, in units of π/d). Radians: c is found with π to double precision, then turned into degrees; a c within 10⁻⁹° of a whole degree (or |θ| × 10⁻¹³°, if larger, for radians) is taken as that degree.
Angles from −10⁹ to 10⁹ in degrees or radians, or −10¹² π to 10¹² π.

Output format. Degrees and radians to 10 significant figures, the unit-circle coordinates to 4 decimals, π forms as "π/3", "2π/9", "0".

Worked examples by hand

225°. In quadrant III: 225° − 180° = 45° = π/4 (0.785398 rad).

5π/3. In quadrant IV: 2π − 5π/3 = π/3 (60°); coterminal 300°.

−π/6. −π/6 + 2π = 11π/6, in quadrant IV: 2π − 11π/6 = π/6; coterminal 330°.

870°. 870° − 2 × 360° = 150°, quadrant II: 180° − 150° = 30° (π/6).

4 radians. 4 is between π (3.14159) and 3π/2 (4.71239), so quadrant III: 4 − π = 0.858407 rad = 49.183118°.

−90°. −90° + 360° = 270°, on the negative y-axis: 270° − 180° = 90° (π/2).

Other questions people ask

What is a reference angle?

The acute angle between the terminal side of an angle and the x-axis, from 0° to 90°. Angles with the same reference angle have the same sine and cosine up to the sign, so it lets you use the first-quadrant values for any angle.

How do I find the reference angle?

First add or subtract 360° until the angle is from 0° to under 360°. Then use its quadrant: in quadrant I the angle is its own reference angle; in II use 180° − angle; in III use angle − 180°; in IV use 360° − angle. 225° is in quadrant III, so its reference angle is 225° − 180° = 45°.

How do I find a reference angle in radians?

The same rules with π for 180° and 2π for 360°: π − t in quadrant II, t − π in III, 2π − t in IV. 5π/3 is in quadrant IV, so its reference angle is 2π − 5π/3 = π/3. Pick × π and type 5/3 to get the exact answer.

What is the reference angle of a negative angle?

Add 360° (or 2π) first. −π/6 + 2π = 11π/6, which is in quadrant IV, so the reference angle is 2π − 11π/6 = π/6.

What is the reference angle of an angle larger than 360°?

Subtract 360° as many times as needed. 870° − 720° = 150°, in quadrant II, so the reference angle is 180° − 150° = 30°.

What is the reference angle of 90° or 180°?

An angle on an axis has no quadrant. The page uses the same rules at the boundaries, so an angle on the x-axis (0°, 180°) has reference angle 0°, and one on the y-axis (90°, 270°) has reference angle 90°.