Supplementary Angles Calculator (180° − θ in degrees and radians)

Understanding the supplementary angle calculator — concept and principle

This calculator finds the supplement of an angle, the angle that adds with it to 180°, in degrees and radians. It also says whether the supplement is acute, right or obtuse, gives the shared sine value and shows the complement when one exists. Use it for geometry homework, parallel-line problems, polygon angles and carpentry cuts.

How it is calculated

• Supplement = 180° − θ (or π − θ in radians).
• Radians = degrees × π ÷ 180.
• sin θ = sin(180° − θ), so both angles share the same sine.
• Complement = 90° − θ, shown only for angles below 90°.

Angles strictly between 0° and 180° have a positive supplement.

The principle

Two angles are supplementary when together they form a straight angle, a half turn of 180°. That happens naturally whenever two angles sit side by side on a straight line. The same relation appears with parallel lines: co-interior angles (same-side interior angles) add to 180°. In a cyclic quadrilateral, opposite angles are supplementary. On the unit circle, θ and 180° − θ are mirror images across the y-axis, so they have the same height (sine) and opposite horizontal positions (cosines of opposite sign).

Worked example

Default inputs: θ = 125°, degrees.

1. Supplement = 180° − 125° = 55°.
2. In radians: 55 × π ÷ 180 ≈ 0.959931.
3. 55° is between 0° and 90°, so the supplement is an acute angle.
4. sin 125° = sin 55° ≈ 0.819152.
5. 125° is larger than 90°, so it has no complement (shown as —).

How to read the result

The supplement of an acute angle is obtuse and the other way round; 90° is its own supplement. If you enter an angle of 180° or more, there is no positive supplement. The shared sine is useful in the law of sines, where an angle and its supplement give the same ratio, which is the source of the "ambiguous case".

Common mistakes

• Confusing supplementary (180°) with complementary (90°).
• Mixing degrees and radians.
• Assuming supplementary angles must be adjacent; any two angles adding to 180° qualify.
• Expecting supplementary angles to have equal cosines; their cosines have opposite signs.

Frequently asked questions

What is the supplement of 90°?

90°, because 90° + 90° = 180°.

Can an angle have a supplement but no complement?

Yes. Any angle between 90° and 180° has a supplement but no positive complement.

Are linear pair angles supplementary?

Yes. Two adjacent angles forming a straight line always add to 180°.

Sources: OpenStax, Contemporary Mathematics (2023), §10.2 Angles · OpenStax, Precalculus 2e (2021), §5.1