Reflex Angle: Definition, Degree Range, and Examples

TL;DR

A reflex angle is any angle greater than 180° and less than 360° - the larger opening on the outside of an ordinary angle. This article defines the reflex angle, shows how to measure one with a standard protractor using the 360°−θ rule, explains why 180° and 360° are excluded, and works through examples from clock hands.

The Larger Angle Hiding On The Other Side Of Every Corner

Every time you draw two rays from a point, you create two angles at once: the small one you meant to draw, and a much larger one wrapping around the other side. That larger opening is the reflex angle, and most people never notice it.

A reflex angle is an angle whose measure is greater than 180° and less than 360°. It is the "outside" companion to a smaller angle: whenever two rays form an ordinary angle ( \theta ) below 180°, the rest of the full turn on the far side is the reflex angle, and the two together make one complete 360° rotation. Reflex angles are one of the six named types in types of angles, sitting just past the straight angle.

By the end, you will be able to name a reflex angle on sight, measure one with an ordinary protractor, and see why 180° and 360° do not qualify.

The Degree Range: Strictly Between 180° And 360°

A reflex angle lives in the open interval between the straight angle and the full turn:

180° < reflex angle < 360°

So 181°, 210°, 270°, and 359° are all reflex angles, while the endpoints are not.

180° is a straight angle, not reflex. Its two rays point in exactly opposite directions and form a straight line, so nothing "wraps around."
360° is a complete angle (a full rotation), not reflex. The ray has returned to its start.

Is 180° a reflex angle? No. A reflex angle must be strictly greater than 180°. At exactly 180°, the figure is a straight line, which is the straight angle, so it falls just below the reflex range.

This places the reflex angle above every angle you meet first. An acute angle is below 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, a straight angle is exactly 180°, and only then does the reflex range begin.

How to Measure A Reflex Angle With A 180° Protractor

A standard protractor only reaches 180°, so you cannot read a reflex angle directly. Use the complement-to-a-full-turn method.

Measure the smaller angle. Place the protractor on the two rays and read the ordinary angle ( \theta ), which is below 180°.
Subtract from 360°. The reflex angle is the rest of the full turn:

[ \text{Reflex angle} = 360° - \theta ]

For example, if the smaller angle reads ( \theta = 120° ), the reflex angle is ( 360° - 120° = 240° ). This works because the ordinary angle and its reflex partner always complete one full rotation together.

Examples of Reflex Angle

Example 1

Classify each angle as acute, right, obtuse, straight, or reflex: 47°, 200°, 180°, 325°.

47° is below 90°, so it is acute.
200° is between 180° and 360°, so it is reflex.
180° is exactly a straight angle, not reflex.
325° is between 180° and 360°, so it is reflex.

Example 2

A student measures an ordinary angle as 110° and reports the reflex angle as 110° too. Spot the error.

110° is below 180°, so it cannot be a reflex angle at all, and that check alone shows the answer is wrong. The reflex angle is the other opening around the same two rays:

[ 360° - 110° = 250° ]

The reflex angle is 250°. Always test the result: if the number is not above 180°, it is not the reflex angle yet.

Example 3

The reflex angle at a vertex is 290°. Find the ordinary (non-reflex) angle at the same vertex.

The two angles complete a full turn:

[ \text{Ordinary angle} = 360° - 290° = 70° ]

Example 4

At 8 o'clock, find the reflex angle between the hour and minute hands of a clock.

The minute hand points at 12, the hour hand at 8. The smaller angle between them spans 4 hour-marks, and each hour-mark is 30°:

[ \text{Smaller angle} = 4 \times 30° = 120° ]

Measured the long way round, the reflex angle is:

[ 360° - 120° = 240° ]

The reflex angle between the hands at 8 o'clock is 240°.

Example 5

Two rays make a right angle. Find the reflex angle on the other side.

A right angle is 90°:

[ \text{Reflex angle} = 360° - 90° = 270° ]

Example 6

A polygon has an inward-pointing corner (a concave vertex). Its interior angle there is measured as 30° on the inside. Explain which angle the interior really uses.

For a concave polygon, the true interior angle at that vertex is the reflex one:

[ \text{Interior (reflex) angle} = 360° - 30° = 330° ]

Where Reflex Angles Earn Their Keep: Turning Past Straight

The reflex angle matters wherever motion or shape carries something more than halfway around a point, not just a little way from a starting line.

Concave shapes. A star, an arrowhead, or an L-shaped room has at least one corner that bends inward.
Rotation and navigation. Describing a move as "250° clockwise" instead of "110° the other way" can matter for which cables or joints reach their limit.

Mistakes to Watch for

Mistake 1: Calling any large-looking angle "reflex."

Where it slips in: Eyeballing an obtuse angle near 170° and labeling it reflex because it looks wide.

Don't do this: Assume "big angle equals reflex."

Mistake 2: Reporting the small protractor reading as the reflex angle

Don't do this: Read 130° off the protractor and write "reflex angle = 130°."

Mistake 3: Adding instead of subtracting from 360°

Don't do this: Write reflex = 360° + ( \theta ).

Key Takeaways

A reflex angle is greater than 180° and less than 360°.
180° (straight) and 360° (complete) are excluded from the reflex range.
Measure a reflex angle as ( 360° - \theta ), where ( \theta ) is the ordinary angle read on a protractor.
Reflex angles appear as the interior angles of concave shapes and as large rotations.

Frequently Asked Questions

What is a reflex angle in simple terms?

An angle bigger than a straight line but smaller than a full turn — that is, between 180° and 360°. It is the large opening on the far side of an ordinary angle.

Can a triangle have a reflex angle?

No. A triangle's three angles sum to 180°, so no single angle can exceed 180°. Reflex interior angles only appear in concave polygons with four or more sides.