Clock angle problems are a type of mathematical problem which involve finding the angle between the hands of an analog clock.

Math problem

Clock angle problems relate two different measurements: angles and time. The angle is typically measured in degrees from the mark of number 12 clockwise. The time is usually based on a 12-hour clock.

A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates through 360° in 60 minutes or 6° per minute.

Equation for the angle of the hour hand

where:

• is the angle in degrees of the hand measured clockwise from the 12

• is the hour.

• is the minutes past the hour.

Equation for the angle of the minute hand

where:

• is the angle in degrees of the hand measured clockwise from the 12 o'clock position.

• is the minute.

Example

The time is 2:20. The angle in degrees of the hour hand is:

The angle in degrees of the minute hand is:

Equation for the angle between the hands

The angle between the hands can be found using the following formula:

where

• is the hour

• is the minute

If the angle is greater than 180 degrees then subtract it from 360 degrees.

Example 1

The time is 2:20.

Example 2

The time is 10:16.

When are the hour and minute hands of a clock superimposed?

The hour and minute hands are superimposed only when their angle is the same.

is an integer in the range 0–11. This gives times of: 0:00, 1:05., 2:10., 3:16., 4:21., 5:27.. 6:32., 7:38., 8:43., 9:49., 10:54., and 12:00. (0. minutes are exactly 27. seconds.)

See also

• Clock position

References

External links

• https://web.archive.org/web/20100615083701/http://delphiforfun.org/Programs/clock_angle.htm

• http://www.ldlewis.com/hospital_clock/ - extensive clock angle analysis

• https://web.archive.org/web/20100608044951/http://www.jimloy.com/puzz/clock1.htm