# 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](/content/math/geometry/types-of-angles/index.html), 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](/content/math/geometry/acute-angle/index.html) is below 90°, a right angle is exactly 90°, an [obtuse angle](/content/math/geometry/obtuse-angle/index.html) 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. 1. **Measure the smaller angle.** Place the protractor on the two rays and read the ordinary angle \( \theta \), which is below 180°. 2. **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.