Clock Angle Formula with Worked Examples

Key Points

• The clock angle formula is $\theta = \lvert 30H - 5.5M \rvert$, where $H$ is the hour and $M$ the minutes past it.

• The 30 comes from the hour hand covering 360 degrees in 12 hours, so each hour mark is 30 degrees apart.

• The 5.5 combines two motions: the minute hand moves 6 degrees per minute while the hour hand creeps forward 0.5 degrees per minute.

• The formula can exceed 180 degrees, in which case the smaller angle between the hands is $360$ minus the result.

• Forgetting that the hour hand moves during the hour is the single biggest source of wrong answers.

What Is the Clock Angle Formula?

The clock angle formula gives the angle between the hour and minute hands at any time:

$$\theta = \lvert 30H - 5.5M \rvert$$

Here $H$ is the hour shown and $M$ is the number of minutes past it. The absolute value makes the answer positive whichever hand is ahead.

Where the 30 and the 5.5 Come From

Each of the two hands has its own constant speed. The minute hand sweeps a full 360 degrees in 60 minutes, so it moves 6 degrees per minute.

The hour hand covers 360 degrees in 12 hours, which is 30 degrees per hour, or $\frac{30}{60} = 0.5$ degrees per minute. That slow creep is what most wrong answers ignore.

Measuring both from the 12 o'clock mark, the minute hand sits at $6M$ degrees and the hour hand at $30H + 0.5M$. Subtracting gives $30H + 0.5M - 6M = 30H - 5.5M$, and the absolute value removes the sign.

Variable Key

Examples of the Clock Angle Formula

Example 1

Find the angle between the hands at 3:00.

$$\theta = \lvert 30(3) - 5.5(0) \rvert = 90$$

Final answer: 90 degrees

Example 2

Find the angle at 3:40, treating the hour hand as fixed on the 3.

If the hour hand stays at the 3 it sits at 90 degrees, and the minute hand at 40 minutes sits at $6 \times 40 = 240$ degrees, giving $240 - 90 = 150$ degrees.

That ignores the hour hand's own movement. By 40 minutes past, it has travelled $0.5 \times 40 = 20$ degrees beyond the 3, so it is at 110 degrees rather than 90.

Using the formula, which accounts for both motions:

$$\theta = \lvert 30(3) - 5.5(40) \rvert = \lvert 90 - 220 \rvert = 130$$

Final answer: 130 degrees, not 150. The 20 degree discrepancy is exactly the hour hand's creep.

Example 3

Find the angle at 9:15.

$$\theta = \lvert 30(9) - 5.5(15) \rvert = \lvert 270 - 82.5 \rvert = 187.5$$

Since this exceeds 180, take the smaller angle:

$$360 - 187.5 = 172.5$$

Final answer: 172.5 degrees

Example 4

Find the angle at 12:30.

$$\theta = \lvert 30(12) - 5.5(30) \rvert = \lvert 360 - 165 \rvert = 195$$ $$360 - 195 = 165$$

Final answer: 165 degrees

Using $H = 12$ rather than 0 keeps the formula consistent, and the subtraction from 360 handles the rest.

Example 5

At what time between 4 and 5 o'clock are the hands together?

Set the angle to zero with $H = 4$:

$$30(4) - 5.5M = 0 \implies M = \frac{120}{5.5} = 21.818\ldots$$

Final answer: about 4:21 and 49 seconds, since $0.818$ of a minute is about 49 seconds.

The hands coincide once in most hours, and the awkward fraction is why clock-coincidence times are rarely whole minutes.

Two Angles, One Pair of Hands

Two hands always make two angles that sum to 360. Unless a question asks for the reflex angle, the expected answer is the smaller one.

Clock Angle, Where Things Go Sideways

Mistake 1: Treating the hour hand as fixed

Where it slips in: The hour hand moves slowly enough to look stationary within an hour.

Don't do this: Place the hour hand exactly on its hour mark for any time past the hour.

The correct way: Add $0.5M$ degrees to its position, which the $5.5$ in the formula already does. At 3:40 that is a 20 degree difference.

Students who sketch the clock and deliberately draw the hour hand past its mark get this right, because the drawing forces the question of how far past. Reading the time as a pair of numbers is what makes the hand feel fixed.

Mistake 2: Keeping an answer above 180 degrees

Where it slips in: The formula returns a valid number and it gets reported as it stands.

Don't do this: Report 187.5 degrees as the angle at 9:15.

The correct way: Subtract from 360 to get 172.5. Clock-angle questions want the smaller of the two angles unless they say otherwise.

Mistake 3: Using 0 for twelve o'clock

Where it slips in: Digital clocks show 12 as 00, so zero looks like the natural value.

Don't do this: $H = 0$ for 12:30.

The correct way: Use $H = 12$. The formula is built on hour marks from 1 to 12, and substituting 0 drops the full turn that makes the subtraction work.

Key Takeaways

• The clock angle formula is $\theta = \lvert 30H - 5.5M \rvert$ degrees.

• The 30 is the hour-mark spacing and the 5.5 is the rate at which the minute hand gains on the hour hand.

• The hour hand moves 0.5 degrees per minute, and ignoring that creep is the dominant error.

• A result above 180 is converted by subtracting from 360, giving the smaller angle.

• To work through angle problems with a teacher, explore a 6th grade math tutor or math tuition.

Practice These Three Before Moving On

Find the angle at 2:20 and at 7:10. Then work out the first time after 8 o'clock when the hands are exactly together.

Want a live Bhanzu trainer to walk through more angle problems? Book a free demo class.

Read More

• Degrees

• Angle Between Two Lines

• Constructing an Angle of 90 Degrees

• Time Formula

• 180 Degrees to Radians