Each hour mark is 30 degrees

Learning outcome

You can find the smaller angle between the hands at an exact hour.

Concepts and reasons

A clock is a circle of 360 degrees, and it has 12 hour marks. One gap is 360 / 12 = 30 degrees. At an exact hour the minute hand is on 12. The hour hand is on that hour mark. The hour hand has moved H times 30 degrees from 12. At 3:00 that is 90 degrees. At 6:00 it is 180. At 8:00 it is 240. The exam wants the smaller angle. If the long way is more than 180, subtract it from 360. At 8:00 the long way is 240, so the smaller angle is 360 - 240 = 120. The short way is 4 gaps, and 4 times 30 is also 120. At 12:00 both hands are on the same mark. The angle is 0.

Worked examples

1. At 3:00 the hour hand is 3 gaps from 12. 3 times 30 = 90. The smaller angle is 90 degrees.
2. At 6:00 the hands are opposite. 6 times 30 = 180.
3. At 10:00 the hour hand is 10 times 30 = 300 degrees from 12. The smaller angle is 360 - 300 = 60 degrees. The short way is 2 gaps.

Common traps

8 times 30 = 240 is the long way at 8:00. The answer is 120. A right angle is 90 degrees. A straight line is 180. Do not call 120 a right angle. 12:00 is 0 degrees, not 360 as a second answer. Both describe the same meeting of the hands.

Practice questions

1. What is the smaller angle at 3:00?
2. What is the smaller angle at 8:00?
3. What is the smaller angle at 4:00?

Answers and explanations

1. 90 degrees. 3 gaps times 30.
2. 120 degrees. 8 gaps is 240, and 360 - 240 = 120. The short way is 4 gaps.
3. 120 degrees. 4 gaps times 30. 120 is already under 180, so it is the smaller angle.

Analogy

The 12 marks are 12 slices of a round clock. Each slice is 30 degrees. At an exact hour you only count the slices between 12 and the hour hand, then keep the shorter count.

Quick reference

• One hour gap = 30 degrees.
• At H o clock the long count is H times 30, with the minute hand on 12.
• If that count is over 180, the smaller angle is 360 minus it.
• 3:00 is 90. 6:00 is 180. 12:00 is 0.