The angle after some minutes

Learning outcome

You can find the smaller angle when the time is hours and minutes, not an exact hour.

Concepts and reasons

The minute hand moves 6 degrees in one minute, because 360 / 60 = 6. The hour hand moves 30 degrees in one hour, so it also moves 0.5 degree in one minute. At H hours and M minutes, the hour hand is at 30H + 0.5M degrees from 12. The minute hand is at 6M degrees from 12. The difference is the absolute value of 30H - 5.5M. The 5.5 is 6 minus 0.5. If that difference is more than 180, the smaller angle is 360 minus the difference.

Worked examples

1. At 2:20, 30 times 2 minus 5.5 times 20 = 60 - 110. The absolute value is 50. 50 is under 180, so the smaller angle is 50 degrees. Check: the hour hand is at 60 + 10 = 70, the minute hand is at 120, and 120 - 70 = 50.
2. At 5:30, 150 - 165. The absolute value is 15 degrees.
3. At 10:10, 300 - 55. The absolute value is 245. 245 is over 180, so the smaller angle is 360 - 245 = 115 degrees.

Common traps

At 2:20 the hour hand is not still on the 2. It has moved 10 degrees past 2. 245 degrees is the long way at 10:10. The exam wants 115. Do not drop the 0.5. A formula that uses only 30H treats the hour hand as frozen.

Practice questions

1. What is the smaller angle at 2:20?
2. What is the smaller angle at 4:40?
3. What is the smaller angle at 10:10?

Answers and explanations

1. 50 degrees. The absolute value of 60 - 110 is 50.
2. 100 degrees. 30 times 4 minus 5.5 times 40 = 120 - 220. The absolute value is 100, and 100 is under 180. Check: hour hand at 140, minute hand at 240.
3. 115 degrees. The absolute value of 300 - 55 is 245, and 360 - 245 = 115.

Analogy

Both hands leave 12, but the minute hand is the faster walker. You mark where each walker stands, then take the shorter arc between those two marks.

Quick reference

• Minute hand: 6 degrees a minute. Hour hand: 0.5 degree a minute.
• Difference = absolute value of 30H - 5.5M.
• If the difference is over 180, smaller angle = 360 minus the difference.
• 2:20 is 50 degrees. 5:30 is 15 degrees. 10:10 is 115 degrees.