What are the types of angles?

Full turn, straight angle and right angle

Vidya makes a full turn of her book’s cover when she folds it right round to write while holding the book: that is 360°. When she opens the book flat on the table, the cover makes a half turn and its two arms lie in a straight line. Such an angle is a straight angle: 180°.

Draw a straight angle AOB along the edge of a sheet of paper and fold the paper so that OB lies exactly on OA. The crease divides the straight angle into two equal angles. Each is a right angle, a quarter of a full turn: 90°. So a straight angle contains two right angles.

A right angle looks like the letter L. Two lines that meet at a right angle are perpendicular to each other. You can find right angles at the corners of windows, doors, books and the blackboard.

Acute and obtuse angles

Compare any angle with a right angle, for example the corner of a sheet of paper:

• An angle less than a right angle, less than a quarter turn, is an acute angle.
• An angle greater than a right angle but less than a straight angle, between a quarter and a half turn, is an obtuse angle.

Acute means sharp and obtuse means blunt, which matches how these angles look: an acute angle is narrow and pointed, an obtuse angle wide and blunt.

Types of angles by measure

Type	Measure	Turn
Acute	more than 0° and less than 90°	less than a quarter turn
Right	exactly 90°	a quarter turn
Obtuse	more than 90° and less than 180°	between a quarter and a half turn
Straight	exactly 180°	a half turn
Reflex	more than 180° and less than 360°	between a half and a full turn
Full turn	exactly 360°	one complete turn

Reflex angles are the large angles on the outside of two arms. If two arms make 102° on one side, the reflex angle on the other side is 360° − 102° = 258°.

Working with angle types

Using a straight angle. B, E and R lie on a straight line, so ∠REB = 180°. If ∠TER = 80°, then ∠BET = 180° − 80° = 100°. If ES is perpendicular to the line, ∠SER = 90° and ∠SET = 90° − 80° = 10°.

Angles in a clock. The 12 numbers split the full turn of 360° into 12 equal parts of 30°. At 1 o’clock the hands make 30° (acute), at 3 o’clock 90° (right), at 4 o’clock 120° (obtuse) and at 6 o’clock 180° (straight).

The Ashoka Chakra. Its 24 equally spaced spokes divide 360° into 24 parts of 15°. The largest acute angle between two spokes is 5 × 15° = 75°, because 6 × 15° = 90° is already a right angle.

A puzzle from the book. An acute angle stays acute when doubled, tripled and quadrupled, but becomes obtuse when multiplied by 5. Four times it must be less than 90°, so it is less than 22.5°; five times it must be more than 90°, so it is more than 18°. In whole degrees it can be 19°, 20°, 21° or 22°.

To find the exact measure of an angle, use a protractor: see how to measure an angle with a protractor. For the whole chapter, see the Chapter 2 notes.