Acute, right, obtuse and reflex angles

At 2:30 the hands of a clock are 105° apart, not 90°. The hour hand has already moved halfway from the 2 to the 3.

Five names worth knowing

Some angles come up so often that they have names. OpenStax lists four of them. A right angle measures exactly 90°, the corner of a book or a tile. An acute angle is any angle smaller than 90°. An obtuse angle is bigger than 90° but smaller than 180°. A straight angle measures exactly 180°: its two sides point in opposite directions and together form a straight line. Angles larger than 180° but smaller than 360° also have a name, reflex angles, and you meet them whenever you measure the outside of a corner.

The words are about ranges, so the boundaries matter. 89° is acute and 91° is obtuse, but 90° is neither: it is right. Likewise 180° is straight, not obtuse and not reflex. Two lines that meet at a right angle are called perpendicular, and a drawing marks this with a little square in the corner instead of writing 90°.

Angles on a clock face

A clock face is a ready-made protractor. Its twelve hour marks split 360° into twelve equal parts, so neighbouring marks are 360 ÷ 12 = 30° apart. At 3:00 the hands are three marks apart, 90°, a right angle. At 4:00 they are 120° apart, obtuse. At 6:00 they make a straight angle.

Times that are not on the hour need one more fact. The minute hand turns 360° in 60 minutes, which is 6° per minute. The hour hand turns only 30° in those 60 minutes, which is 0.5° per minute, so it creeps forward as the hour goes on. At 2:30 the hour hand has moved 2 × 30 + 30 × 0.5 = 75° from the 12, and the minute hand is at 30 × 6 = 180°. The angle between them is 180 − 75 = 105°. If the difference ever comes out above 180°, subtract it from 360° to get the smaller angle between the hands.

Checking a right angle without a protractor

Builders laying out a wall or a garden bed need a true right angle over several metres, far bigger than any protractor. A triangle with sides 3, 4 and 5 does the job, because 3² + 4² = 9 + 16 = 25 = 5², and by the converse of the Pythagorean theorem a triangle whose sides fit that rule has a right angle between the two shorter sides. Mark 3 m along one line and 4 m along the other; if the straight distance between the two marks is exactly 5 m, the corner is square. Any multiple works too, such as 6, 8 and 10.

Turning angles can also go beyond one revolution. A wheel that turns twice has turned through 720°, and a skater who spins one and a half times has turned 540°. For the shape of a corner, though, only the part less than 360° matters, which is why a protractor stops at one turn.