几何角度计算器

The Geometry Angle Calculator takes any single angle and returns its supplementary angle (the one that pairs with it to make 180°), its complementary angle (the pair that makes 90°), its reflex angle (the pair that makes a full 360°), and its classification (acute, right, obtuse, straight, or reflex). This tutorial explains what each of these relationships means, where they appear in geometry problems, and how to avoid the most common student mistakes around them.

Angle classification — naming an angle by its size

Every angle is classified by its measurement:

Supplementary angles — pair to 180°

Two angles are supplementary if they sum to 180°.

Formula: supplement of θ = 180° − θ.

If θ = 50°, its supplement is 130°. If θ = 130°, its supplement is 50°. The relationship is mutual.

Where supplementary angles appear in geometry:

• Linear pair: when two rays form a straight line, the angles on either side of any point on that line sum to 180°.
• Co-interior angles (same-side interior) when a transversal crosses two parallel lines.
• Opposite angles in a cyclic quadrilateral: a quadrilateral inscribed in a circle has opposite angles summing to 180°.
• Adjacent angles in a parallelogram: each consecutive pair sums to 180°.

Complementary angles — pair to 90°

Two angles are complementary if they sum to 90°.

Formula: complement of θ = 90° − θ.

If θ = 30°, its complement is 60°. If θ = 60°, its complement is 30°.

Important constraint: only acute angles have a complement. The complement of 100° is "−10°" — a meaningless negative angle. The calculator reports "no complementary angle" when the input exceeds 90°.

Where complementary angles appear:

• The two acute angles of a right triangle: they sum to 90° (since all three angles total 180° and one is 90°).
• Adjacent angles forming a right angle: two angles sharing a vertex and side that together make 90°.
• Trigonometric identity: sin(θ) = cos(90° − θ). This identity is the reason cosine is named "co-sine" (the sine of the complementary angle).

Reflex angles — pair to 360°

For any angle θ less than 360°, the reflex angle is 360° − θ.

Formula: reflex of θ = 360° − θ.

If θ = 70°, the reflex angle is 290°. If θ = 290°, the reflex angle is 70°. They are the two ways to measure the "opening" between two rays — one going around the short way, one going around the long way.

Reflex angles appear when a concave polygon (like a dart or arrowhead shape) has one vertex where the interior angle exceeds 180°. In such cases the "interior" angle is reflex.

Worked examples

Example 1 — Acute input: θ = 35°.

• Class: Acute (less than 90°)
• Supplement: 180° − 35° = 145°
• Complement: 90° − 35° = 55°
• Reflex: 360° − 35° = 325°

Example 2 — Obtuse input: θ = 150°.

• Class: Obtuse
• Supplement: 180° − 150° = 30°
• Complement: undefined (θ > 90°)
• Reflex: 360° − 150° = 210°

Example 3 — Right angle: θ = 90°.

• Class: Right
• Supplement: 90° (its own supplement!)
• Complement: 0° (only the zero angle is its own complement; 90° is the boundary case)
• Reflex: 270°

Notation conventions

Several notations exist for angles:

• θ (theta): the most common Greek letter for a generic angle.
• ∠ABC: the angle at vertex B, formed by rays BA and BC.
• m∠ABC: "the measure of ∠ABC", typically in degrees.
• °: the degree symbol. 90° = "ninety degrees".
• rad: radians. 1 radian ≈ 57.296°. π radians = 180°.

Our calculator uses degrees. If your problem is in radians, convert: degrees = radians × 180/π.

Acute / obtuse and triangle classification

Triangles are classified by their largest angle:

• Acute triangle: all three angles less than 90°.
• Right triangle: one angle exactly 90°.
• Obtuse triangle: one angle greater than 90°.

The 180° angle sum rule means at most ONE angle in any triangle can be obtuse — if two were, they'd sum to more than 180° before the third is added.

Common mistakes

• Confusing supplementary and complementary. Memorize: Supplementary = Straight line (180°). Complementary = Corner (90°). The mnemonic ties each word to the larger associated concept.
• Computing complement of an obtuse angle. The complement is undefined when θ > 90°. The calculator returns nothing rather than a negative number.
• Calling a 90° angle "acute". A right angle is EXACTLY 90° — neither acute (which is strictly less than 90°) nor obtuse. The boundary case has its own name.
• Adding two reflex angles. Two reflex angles can't both be interior angles of the same convex polygon — they sum to more than 360°, which violates the polygon interior-angle sums.
• Degrees vs radians mode confusion. Common on calculators with both modes. sin(30°) ≈ 0.5; sin(30 rad) ≈ −0.988. Make sure mode matches the unit your problem uses.