A reflex angle is any angle greater than 180 degrees and less than 360 degrees. To measure one with an ordinary 180-degree protractor, first measure the smaller angle between the two arms, then subtract it from 360. For example, if the inside angle is 120 degrees, the reflex angle is 240 degrees.
Most angles you meet in a geometry class open less than a straight line. This angle is the odd one out, because it wraps around the “back” of the vertex. As a result, it looks bent backward, and a standard half-circle protractor cannot read it in one step.
Still, every reflex angle comes with a partner. After all, two arms that meet at a point always make two angles: a smaller one on one side and a larger one on the other. Together, they add up to a full turn of 360 degrees. So if you know one, you also know the other.
In short: measure the small side, subtract it from 360, and you have the reflex angle.
The degree itself is a fixed unit. For instance, the US National Institute of Standards and Technology lists the degree in its SI unit guide (SP 811) as exactly pi/180 radian, which is why a full turn is 360 degrees, or 2 pi radians. Similarly, the Math Is Fun definition describes it as more than 180 degrees but less than 360 degrees.
In the sections below, you will also find a worked-example table, a step-by-step method, everyday examples, and the tools that make the job easier.
Convert an Angle to Slope and Roof Pitch
This converter turns an angle in degrees into percent grade and roof pitch (rise per 12 inches of run). However, slope only makes sense between -90 and 90 degrees, so enter the small, inside angle here. Then, for the reflex side, use 360 minus that angle, as shown in the table further down.
- This kind of angle measures more than 180 degrees and less than 360 degrees.
- Exactly 180 degrees is a straight angle, and exactly 360 degrees is a full turn.
- Reflex angle = 360 degrees minus the smaller (inside) angle.
- The two angles add up to 360 degrees and are called explementary or conjugate angles.
- In radians, a reflex angle lies between pi (about 3.1416) and 2 pi (about 6.2832).
- A 360-degree protractor or a digital angle finder can read many of them directly.
- Clock hands and a pizza with a slice missing are easy real-world examples.
Reflex Angle Formula With Worked Examples
The formula is short: reflex angle = 360 – inside angle. Because the two angles share the same arms and the same vertex, they always fill a complete circle. Therefore, the smaller the inside angle, the larger the angle on the other side.
| Inside angle (measured) | Calculation | Result | Result in radians |
|---|---|---|---|
| 30° | 360 – 30 | 330° | 5.760 rad |
| 45° | 360 – 45 | 315° | 5.498 rad |
| 60° | 360 – 60 | 300° | 5.236 rad |
| 90° | 360 – 90 | 270° | 4.712 rad |
| 120° | 360 – 120 | 240° | 4.189 rad |
| 135° | 360 – 135 | 225° | 3.927 rad |
| 150° | 360 – 150 | 210° | 3.665 rad |
| 170° | 360 – 170 | 190° | 3.316 rad |
| 179° | 360 – 179 | 181° | 3.159 rad |
You can also run the formula backward. For example, if a drawing labels the outside angle as 290 degrees, the inside angle is 360 – 290 = 70 degrees. In addition, this check helps when you want to confirm a printed value with a protractor.
Reflex Angle vs Other Types of Angles
In general, angles are grouped by size. Here is how this one fits with the rest. For more detail on each group, see our guide to the different types of angles, from acute to obtuse.
| Angle type | Size | Quick example |
|---|---|---|
| Zero angle | 0° | Two clock hands on top of each other at 12:00 |
| Acute | more than 0°, less than 90° | The tip of a pizza slice |
| Right | exactly 90° | The corner of a sheet of paper |
| Obtuse | more than 90°, less than 180° | A reclined laptop screen |
| Straight | exactly 180° | Clock hands at 6:00 |
| Reflex | more than 180°, less than 360° | The outside of the clock hands at 4:00 (240°) |
| Full angle (full turn) | exactly 360° | One complete rotation |
The table also shows a common mix-up. Supplementary angles add up to 180 degrees, so they can never include an angle over 180 degrees. Instead, an angle and its reflex partner add up to 360 degrees. That is why mathematicians call such a pair explementary or conjugate angles.
How to Measure a Reflex Angle With a Protractor
A standard school protractor is a half circle marked from 0 to 180 degrees. So it cannot reach past a straight line. Instead, you measure the inside angle and let arithmetic do the rest. Our step-by-step guide on how to measure angles using a protractor covers the basic placement in more detail.
- Find the vertex. Mark the point where the two arms meet. If the arms are short, extend them with a ruler so they cross the protractor scale.
- Place the protractor. Put the center hole or crosshair exactly on the vertex, and line up the zero line with one arm.
- Read the inside angle. Follow the other arm to the scale. Then read the number on the scale that starts at 0 on your first arm.
- Subtract from 360. Next, take 360 minus your reading. For example, a reading of 75 degrees gives a reflex angle of 285 degrees.
- Check the answer. Finally, make sure the result is between 180 and 360. If it is not, you probably read the wrong scale.
There is a second method, too. First, extend one arm straight through the vertex to make a 180-degree line. Then measure the extra angle beyond that line and add 180. For instance, an extra 60 degrees gives 180 + 60 = 240 degrees. Both methods give the same result, so you can use one to check the other.
Reflex Angle Examples in Everyday Life
Once you know what to look for, reflex angles show up all over the place. For example, here are a few you can check yourself.
Clock hands
To begin with, a clock face is a circle of 360 degrees, and each hour mark is 30 degrees apart. So at 4:00, the hands are 120 degrees apart on the short side. The outside angle, therefore, is 240 degrees. Similarly, at 10:00 the inside angle is 60 degrees and the outside angle is 300 degrees. At 6:00, however, the hands form a straight angle of exactly 180 degrees, so there is no reflex angle.
Pizza with slices missing
Cut a round pizza into 8 equal slices, and each tip is 360 / 8 = 45 degrees. Take one slice away, and the remaining pizza forms a reflex angle of 315 degrees at the center. Likewise, after two slices are gone, the rest of the pizza spans 270 degrees.
Other places to spot a reflex angle
- A pie chart. When one wedge is small, the rest of the chart forms a reflex angle.
- A door swing. Measured around the far side, a door opened 90 degrees leaves a 270-degree reflex angle.
- Star and arrow shapes. The inward-pointing corners of a star are larger than 180 degrees inside the shape.
- Outside corners of a room. On the hallway side, a wall corner that juts out measures about 270 degrees.
Recommended Tools for Measuring a Reflex Angle
A basic 180-degree protractor works fine with the subtract-from-360 method. However, a full-circle protractor saves a step, and a digital angle finder helps with real objects such as boards, frames and corners. Based on manufacturer specs and buyer reports, these are well-reviewed options. First, look for clear, high-contrast markings. Also, for digital tools, check the measuring range in the manual, because not every model reads all the way to 360 degrees.
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If you want to compare more models, our roundup of the best protractors for school and the shop goes further.
Which Measuring Tool Works Best?
| Tool | Reads past 180 degrees directly? | Best for |
|---|---|---|
| 180-degree protractor | No; measure the inside angle and subtract from 360 | Homework and drawings |
| 360-degree protractor | Yes | Drafting, pie charts, circle work |
| Digital angle finder | Often, within the range listed in the manual | Woodwork, trim, metal frames |
| Compass and ruler | No; used to construct angles, not read them | Geometry constructions |
Do and Don’t for Measuring Big Angles
- Put the protractor center exactly on the vertex.
- Read the scale that starts at 0 on your first arm.
- Estimate the size before you calculate.
- Label which side of the angle you mean in a drawing.
- Double-check with the “180 plus extra” method.
- Confuse supplementary (180) with explementary (360).
- Call exactly 180 or exactly 360 degrees a reflex angle.
- Subtract from 180 when you mean 360.
- Trust a reading taken from a thick, curved pencil line.
- Assume every digital tool covers a full circle.
Honest Limits of Hand-Measured Angles
Any angle you measure by hand carries some error. For example, a school protractor is marked in 1-degree steps, so a careful reading is usually good to about half a degree. Moreover, thick pencil lines and a slightly misplaced vertex can add another degree or two. Subtracting from 360 does not make that error bigger or smaller; it simply carries over to the reflex value. In addition, cheap plastic protractors can be printed slightly off-center. So for drafting or shop work, check your tool against a known 90-degree square first.
Digital angle finders have their own limits as well. For instance, their stated accuracy applies only within the listed range, and only after you zero or calibrate them as the manual describes. In short, the formula is exact, but your reading is only as good as the tool and your setup.
When to Call a Professional
For homework, crafts and most DIY projects, a protractor and the 360 rule are all you need. However, some jobs depend on precise angles. For example, these include roof framing, stair stringers, structural steel and machined parts. In those cases, ask a licensed contractor, surveyor or machinist, and use calibrated tools. Likewise, if a part must meet a tolerance on a drawing, a calibration lab can verify your angle gauges.
Reflex Angle FAQs
What is a reflex angle?
A reflex angle is an angle that measures more than 180 degrees but less than 360 degrees. It is the larger of the two angles formed by two arms that meet at a point.
How do you find a reflex angle from an inside angle?
Subtract the inside angle from 360 degrees. For example, an inside angle of 110 degrees gives an outside angle of 250 degrees.
Is 180 degrees a reflex angle?
No. Exactly 180 degrees is a straight angle. The reflex type must be more than 180 degrees.
Is 360 degrees a reflex angle?
No. Exactly 360 degrees is a full angle, or one complete turn. The reflex range stops just short of 360 degrees.
Can a triangle have an angle over 180 degrees?
No. The three angles of a flat triangle add up to 180 degrees, so none of them can be larger than 180. Shapes with an inward corner, such as stars and arrowheads, can have one.
What is a reflex angle in radians?
A reflex angle lies between pi and 2 pi radians, or about 3.1416 to 6.2832 rad. For example, 270 degrees equals about 4.712 rad.
What are angles that add up to 360 degrees called?
They are called explementary or conjugate angles. An angle and its reflex partner around the same vertex are always explementary.
Can you measure a reflex angle with a 180-degree protractor?
Yes, in two steps. First, measure the smaller inside angle. Then subtract that reading from 360 degrees.
What is the outside angle of a clock at 4:00?
At 4:00, the hands are 120 degrees apart on the short side. So the angle on the outside is 240 degrees.
How many angles over 180 degrees can a polygon have?
A concave polygon has at least one interior angle larger than 180 degrees. A polygon with n sides can have up to n – 3 of them, so a quadrilateral can have only one.
Reflex Angle: The Bottom Line
To sum up, a reflex angle is any angle between 180 and 360 degrees. Because it always pairs with a smaller angle to make a full 360-degree turn, you can find it with simple subtraction: measure the inside angle, then take 360 minus that number.
So whether you are checking a clock face, a slice of pizza or a corner on a drawing, the method stays the same. Finally, if you measure these angles often, a full-circle protractor or a digital angle finder can read many of them directly and save you a step.
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