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:
$$180° < \text{reflex angle} < 360°$$
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.
A natural first move is to read the protractor once and call that number the reflex angle. But $110°$ 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:
$$\text{Ordinary angle} = 360° - 290° = 70°$$
The ordinary angle is $70°$, which is acute, and $70° + 290° = 360°$ confirms it.
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 $90°$:
$$\text{Reflex angle} = 360° - 90° = 270°$$
The reflex angle is $270°$, three-quarters of a full turn.
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 (non-convex) polygon, the interior turns past straight at a dent, so the true interior angle at that vertex is the reflex one, not the small $30°$ opening on the outside.
$$\text{Interior (reflex) angle} = 360° - 30° = 330°$$
The interior angle at that vertex is $330°$, which is why a shape with any reflex interior angle is called concave rather than convex.
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. The true interior angle at such a corner is a reflex angle above $180°$, which is precisely what makes the shape non-convex.
Rotation and navigation. A dancer, a robot arm, or a camera gimbal turning $250°$ has swept a reflex angle. Describing the move as "$250°$ clockwise" instead of "$110°$ the other way" can matter for which cables or joints reach their limit.
Why we bother naming it. The destination is unambiguous direction. Two rays always define two angles; naming the reflex one lets us say exactly which way and how far around we mean, with no confusion between the short route and the long route.
The clock example above is the everyday case: a reflex angle is simply the long way round the dial, and it is documented in every treatment of how clock hands sweep angles across the day.
Mistakes to Watch for
Mistake 1: Calling any large-looking angle "reflex"
Where it slips in: Eyeballing an obtuse angle near $170°$ and labelling it reflex because it looks wide.
Don't do this: Assume "big angle equals reflex."
The correct way: Reflex means strictly greater than $180°$. Anything from $90°$ to $180°$ is obtuse, and exactly $180°$ is straight. Check the number against the $180°$ boundary before naming it, rather than judging by how wide the opening looks.
Mistake 2: Reporting the small protractor reading as the reflex angle
Where it slips in: Measuring with a $180°$ protractor and forgetting the subtraction step.
Don't do this: Read $130°$ off the protractor and write "reflex angle $= 130°$."
The correct way: The protractor gives the ordinary angle. The reflex angle is $360° - \theta$, so $360° - 130° = 230°$. The rusher who reads once and stops skips the step that actually produces the reflex measure.
Mistake 3: Adding instead of subtracting from $360°$
Where it slips in: Recalling that the two angles relate to $360°$ but misremembering how.
Don't do this: Write reflex $= 360° + \theta$, producing an impossible answer above $360°$.
The correct way: The ordinary angle and reflex angle add to $360°$, so the reflex angle is $360° - \theta$. If your answer exceeds $360°$, that impossible value is the flag to switch the operation.
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.
An ordinary angle and its reflex angle always add to $360°$.
Reflex angles appear as the interior angles of concave shapes and as large rotations.
A practical next step
Practice these problems to solidify your understanding. For each, first decide whether the given angle is already reflex, then apply $360° - \theta$ only when you need the companion.
Find the reflex angle for an ordinary angle of $75°$. (Answer to Question 1: $360° - 75° = 285°$.)
Classify $359°$, $180°$, and $181°$. (Answer to Question 2: $359°$ reflex, $180°$ straight, $181°$ reflex.)
To work through angle types with a teacher, explore Bhanzu's geometry tutor, our middle school math tutor sessions, or math classes online. To see a trainer measure reflex angles live, you can book a free demo class.
Read More
Sum of angles in a polygon - how reflex interior angles enter the angle-sum count of concave shapes.
Coterminal angles - angles that share a terminal side once you rotate past $360°$.
Angle addition postulate - how an ordinary angle and its reflex partner combine to $360°$.
Reference angle - the acute angle used to measure large rotations.
Linear pair of angles - the $180°$ straight-angle pairing next to the reflex range.
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