Mathematics
Reflex Angles: Bigger Than a Straight Line
A reflex angle is bigger than 180° and smaller than 360°. Where it sits among acute, right and obtuse angles, and how to measure one with a 180° protractor.
Reflex angle
An angle bigger than 180° but smaller than 360°: more than a straight line, less than a complete turn.
- Where students meet it
- Grade 4 mathematics, when angles are named and measured, and again whenever interior angles of a dart or arrowhead shape come up.
The short answer
A reflex angle is any angle bigger than 180° and smaller than 360° — more than a straight line but less than a full turn. To measure one with an ordinary 180° protractor, measure the other angle at that point and subtract your answer from 360°.
An example
Clock hands at 8 o'clock: 120° one way, 360° − 120° = 240° the other
The twelve hour marks share 360° between them, so each gap is 30°. At 8 o'clock the hands are four gaps apart the short way round, which is 120°, an obtuse angle. Going the long way round the same pair of hands makes a reflex angle of 240°. Both are angles between the same two hands.
Where it sits among the other names
Angles are named by size, and reflex is the largest of the ordinary categories. Every angle you meet at school falls into one of these:
The boundaries are exact. An angle of precisely 180° is a straight angle, not a reflex one, and 360° is a full turn rather than a reflex angle. Reflex means strictly in between.
- Acute — less than 90°, a sharp corner
- Right angle — exactly 90°, marked with a small square
- Obtuse — between 90° and 180°
- Straight angle — exactly 180°, a straight line
- Reflex — between 180° and 360°
- Full turn — exactly 360°, all the way round
Measuring one with a 180° protractor
A school protractor stops at 180°, so a reflex angle cannot be read off directly. Measure the other angle at the same point — the one on the inside, which will be less than 180° — and subtract it from 360°.
If the inner angle measures 110°, the reflex angle is 250°. It is worth writing the subtraction down rather than doing it in your head, because the usual error is giving the inner angle as the answer, and there is nothing in a measurement of 110° to show that anything has gone wrong.
Check the answer against the drawing before moving on. A reflex angle has to come out larger than 180°, so anything smaller means the subtraction from 360° was forgotten.
Where reflex angles actually turn up
Any shape with a dent in it has one. In an arrowhead or dart, three of the four interior angles are ordinary and the fourth, at the point pushed inwards, is reflex. The four still add to 360°, which only works because that one angle is counted as more than 180°.
They also appear whenever a turn is described. A person spinning three-quarters of the way round has turned through 270°, and a compass direction of 300° is a reflex angle measured clockwise from north.
Common questions
Is 180° a reflex angle?
No. An angle of exactly 180° is a straight angle — the two arms lie in a straight line. A reflex angle has to be strictly greater than 180°, so 181° qualifies and 180° does not. The same applies at the top of the range: 360° is a full turn, not a reflex angle.
Can a triangle have a reflex angle?
No. The three angles of a triangle add to 180°, so a single angle of more than 180° would leave the other two adding to a negative amount, which is impossible. Triangles can be acute, right-angled or obtuse, but never reflex.
Can a quadrilateral have one?
Yes, and the shape then looks like an arrowhead or a dart. Because the four interior angles of any quadrilateral add to 360°, one angle of, say, 200° simply means the other three share the remaining 160°. Such a shape is called concave.
Why does my protractor only go up to 180°?
Because a semicircular protractor is built to measure the angles that most questions ask for, and any reflex angle can be reached from its partner. Measure the smaller angle at the point and subtract from 360°. Full 360° circular protractors do exist, but they are rarely used in schools.
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