How often the hands meet or make a right angle

Learning outcome

You can count how often the hands meet, make a right angle, or stand opposite in 12 hours and in 24 hours.

Concepts and reasons

In 12 hours the hour hand goes once around the clock, 360 degrees. The minute hand goes around 12 times, 12 times 360 = 4320 degrees. So in 12 hours the minute hand gains 4320 - 360 = 3960 degrees on the hour hand. The hands meet when that gain is a full round of 360 degrees. 3960 / 360 = 11. They meet 11 times in 12 hours, not 12. A right angle is a gap of 90 degrees, and the other way around is a gap of 270. That is two gaps in each full round of gain. 11 times 2 = 22. They make a right angle 22 times in 12 hours, not 24. They stand opposite when the gap is 180 degrees. That is one gap in each full round, so 11 times in 12 hours. In 24 hours the same clock face runs twice, so each count doubles. They meet 22 times, make a right angle 44 times, and stand opposite 22 times.

Worked examples

1. Meetings in 12 hours: 3960 / 360 = 11.
2. Right angles in 12 hours: two per full round, so 22.
3. Opposite in 24 hours: 11 in 12 hours, and the day doubles it, so 22.

Common traps

They do not meet 12 times in 12 hours. The gain is 3960 degrees, which is 11 full rounds. Two right angles in each of 12 hours would be 24. The gain is not 12 separate hours of 360. It is 3960 degrees, so the count is 22. A day in these questions is 24 hours. Do not leave the 12-hour count as the answer for a day.

Practice questions

1. How many times do the hands meet in 12 hours?
2. How many times do they make a right angle in 12 hours?
3. How many times do they stand opposite in 24 hours?

Answers and explanations

1. 11. The minute hand gains 3960 degrees, and 3960 / 360 = 11.
2. 22. A right angle uses two gaps in each full round of 360 degrees.
3. 22. They stand opposite 11 times in 12 hours, and 24 hours doubles that count.

Analogy

Think of the minute hand as a faster runner on a round track. In 12 hours it gains 11 full laps on the hour hand. Each gained lap gives one meeting, two right angles, and one time when they are opposite.

Quick reference

• In 12 hours the minute hand gains 3960 degrees.
• Meet: 11 times in 12 hours, 22 times in 24 hours.
• Right angle: 22 times in 12 hours, 44 times in 24 hours.
• Opposite: 11 times in 12 hours, 22 times in 24 hours.