Can an angle be exactly 180° or exactly 360° and still be a reflex angle?

No. An angle of exactly 180° is called a straight angle, and an angle of exactly 360° is a full rotation (or complete angle). A reflex angle must be strictly between 180° and 360°.

Reflex Angle — Definition, Formula & Examples

A reflex angle is an angle that measures more than 180° but less than 360°. It is the "larger" angle you get when you go the long way around from one ray to another.

Given two rays sharing a common endpoint, the reflex angle is the angular measure of the rotation from one ray to the other taken in the direction that exceeds a straight angle (180°). Its measure $\theta$ satisfies $180° < \theta < 360°$ .

Key Formula

\theta_{\text{reflex}} = 360° - \theta_{\text{non-reflex}}

Where:

$\theta_{\text{reflex}}$ = The reflex angle (greater than 180°)
$\theta_{\text{non-reflex}}$ = The ordinary (non-reflex) angle between the same two rays (less than 180°)

How It Works

Every pair of rays that form a non-straight angle actually creates two angles: one smaller and one larger. The smaller angle is the one you usually measure (acute, right, or obtuse), while the larger one is the reflex angle. To find the reflex angle, subtract the smaller angle from 360°. For example, if the ordinary angle between two rays is 110°, the reflex angle on the other side is

360° - 110° = 250°

. You can identify a reflex angle visually because it sweeps through more than half a full turn.

Worked Example

Problem: Two rays form an angle of 75°. What is the reflex angle between the same two rays?

Step 1: Identify the non-reflex angle between the two rays.

\theta_{\text{non-reflex}} = 75°

Step 2: Subtract the non-reflex angle from 360° to find the reflex angle.

\theta_{\text{reflex}} = 360° - 75° = 285°

Step 3: Verify the result is between 180° and 360°. Since 180° < 285° < 360°, this is indeed a reflex angle.

Answer: The reflex angle is 285°.

Another Example

Problem: A clock shows 10 o'clock. The minute hand points at 12 and the hour hand points at 10. What is the reflex angle measured clockwise from the minute hand to the hour hand?

Step 1: Find the smaller angle between the two hands. From 12 to 10 going counterclockwise (the short way) is 2 hour positions. Each hour position spans 30°.

2 \times 30° = 60°

Step 2: The reflex angle is the long way around the clock face.

360° - 60° = 300°

Answer: The reflex angle from the minute hand to the hour hand (going clockwise the long way) is 300°.

Visualization

Why It Matters

Reflex angles appear frequently in geometry courses when working with polygons that have interior angles greater than 180° (concave polygons). They also matter in robotics and navigation, where a motor or vehicle may need to rotate more than a half turn. Understanding reflex angles helps you describe any rotation, not just ones under 180°.

Common Mistakes

Mistake: Confusing a reflex angle with an obtuse angle.

Correction: An obtuse angle is between 90° and 180°. A reflex angle is between 180° and 360°. They are different categories — check whether the angle exceeds a straight line (180°).

Mistake: Forgetting that two rays always create two angles that sum to 360°.

Correction: When asked for the reflex angle, subtract the given non-reflex angle from 360°. Do not subtract from 180°.

Related Terms

Angle — General term that includes reflex angles
Acute Angle — Angle less than 90°, contrasts with reflex
Adjacent Angles — Two angles sharing a side can form a reflex pair
Arm of an Angle — The rays that form the reflex angle
Angle Bisector — Can bisect a reflex angle into two parts
Adjacent — Describes objects sharing a boundary or vertex