Classifying Angles
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- Classifying Angles - Video - Naming and Classifying Angles Watch
Theory
Angles are classified by their size. Compare the angle with the benchmarks: a right angle (\(90^\circ\)) and a straight angle (\(180^\circ\)). This Year 7 maths page describes the six types of angle, shows how to read the arc on a diagram and works through classifying, reflex angles and reasoning.
An acute angle is more than \(0^\circ\) and less than \(90^\circ\). A right angle is exactly \(90^\circ\). An obtuse angle is more than \(90^\circ\) and less than \(180^\circ\).
A straight angle is exactly \(180^\circ\). A reflex angle is more than \(180^\circ\) and less than \(360^\circ\). A revolution is exactly \(360^\circ\), one full turn.
From a diagram, look at which side the arc is on. The same two arms make an angle between them and a reflex angle outside them; together they make a revolution.
The six types of angle:
| Type | Size | Looks like |
|---|---|---|
| acute | more than \(0^\circ\), less than \(90^\circ\) | narrower than a square corner |
| right | exactly \(90^\circ\) | a square corner, with a right-angle mark |
| obtuse | more than \(90^\circ\), less than \(180^\circ\) | wider than a square corner |
| straight | exactly \(180^\circ\) | the arms form one straight line |
| reflex | more than \(180^\circ\), less than \(360^\circ\) | the arc goes the long way round |
| revolution | exactly \(360^\circ\) | one full turn |
How to classify an angle
- Look at the arc: between the arms, or the long way round?
- Compare the size with \(90^\circ\) and \(180^\circ\).
- Name the type: acute, right, obtuse, straight, reflex or revolution.
(i) \(70^\circ\)
(ii) \(210^\circ\)
(iii) \(100^\circ\)
(iv) \(315^\circ\)
Compare each size with \(90^\circ\) and \(180^\circ\). \(70^\circ\) is between \(0^\circ\) and \(90^\circ\). \(100^\circ\) is between \(90^\circ\) and \(180^\circ\). \(210^\circ\) and \(315^\circ\) are between \(180^\circ\) and \(360^\circ\).
(i) acute
(ii) reflex
(iii) obtuse
(iv) reflex
| \(360 - 120\) | \(=\) | \(240\) |
(i) The arc is between the arms: obtuse, \(120^\circ\).
(ii) The arc goes the long way round: reflex, \(240^\circ\).
(i) \(A\)
(ii) \(D\)
At \(D\) the corner points into the room, so the inside angle goes the long way round. The outside part is a right angle.
| \(360 - 90\) | \(=\) | \(270\) |
(i) right (\(90^\circ\))
(ii) reflex (\(270^\circ\))
(i) \(2\) o’clock
(ii) \(4\) o’clock
(iii) \(9\) o’clock
| \(2 \times 30\) | \(=\) | \(60\) |
| \(4 \times 30\) | \(=\) | \(120\) |
| \(3 \times 30\) | \(=\) | \(90\) |
(i) \(60^\circ\): acute
(ii) \(120^\circ\): obtuse
(iii) \(90^\circ\): right
(ii) How many different reflex angles have a whole number of degrees?
| \(90 + 35\) | \(=\) | \(125\) |
| \(359 - 181 + 1\) | \(=\) | \(179\) |
(i) \(125^\circ\): obtuse
(ii) Reflex sizes run from \(181^\circ\) to \(359^\circ\): \(179\) angles.
Try an example: \(95^\circ + 100^\circ = 195^\circ\), reflex. But one example is not enough. For any two obtuse angles, each is more than \(90^\circ\), so the sum is more than \(180^\circ\); each is less than \(180^\circ\), so the sum is less than \(360^\circ\).
Yes: the sum is always between \(180^\circ\) and \(360^\circ\), so it is always reflex.
Common pitfalls
Frequently asked questions
What is an acute angle?
An angle less than \(90^\circ\).
What is an obtuse angle?
An angle more than \(90^\circ\) and less than \(180^\circ\).
What is a reflex angle?
An angle more than \(180^\circ\) and less than \(360^\circ\).
What is a revolution?
One full turn, \(360^\circ\).
How do you find a reflex angle from the angle between the arms?
Subtract it from \(360^\circ\). For \(120^\circ\), the reflex angle is \(240^\circ\).