Clock Angle Calculator

At 3:20 the angle between the hour and minute hands is 20°

At 3:20 the minute hand points at the 4, which is 120° clockwise from the 12. The hour hand has not stayed at its 3:00 position of 90°. It has moved 10° further in those 20 minutes and sits at 100°. The smaller angle between the hands is therefore 120° − 100° = 20°, and the larger angle, measured the other way round the dial, is 360° − 20° = 340°. If you leave the hour hand on the 3 you get 30° instead, which is the most common mistake in clock angle problems.

Enter an hour and a minute in the calculator above and it shows the smaller and larger angle, the position of each hand and the type of angle (acute, right, obtuse and so on). On the clock drawing, a dotted line shows how far the hour hand has moved from its on-the-hour position, and you can drag the handles at the hand tips to change the time. To work backwards and find when the hands overlap or form a right angle, choose “Angle → time”. It solves exactly, so answers such as 2:10 10/11 appear as fractions.

How to use it

1. Time → angle: enter the hour (0–23) and the minute. You can use 24-hour input such as 15, and if you add seconds the movement of the hands during those seconds is included. The up and down arrow keys change the hour, minute and second boxes by 1, and one more step up from 59 minutes goes to minute 0 of the next hour.
2. Clock drawing: drag the small handle at the tip of the minute hand to change the time in whole minutes, or the thick handle at the tip of the hour hand to move it to the matching time. With a handle selected, the arrow keys move the minute hand by 1 minute and the hour hand by 1 hour. The orange wedge is the smaller angle between the hands.
3. Next overlap, right angle, straight line: below the result you see the next time after the current one when the hands overlap, form a right angle or form a straight line. Pressing one switches to the “Angle → time” view with the working for that time.
4. Angle → time: choose an angle between the hands (0–360°) and a search range, either all 12 hours or a single hour such as “between 2 and 3”. Pressing a time found updates the clock and the working. An angle over 180° is searched as the equivalent smaller angle (360° minus the value).
5. Now: follows this device's clock every second so you can watch the angle change. Press it again, or edit a box, to stop.
6. Graph: the graph at the bottom shows how the smaller angle rises and falls over 12 hours from 12:00. Pressing the graph calculates that time. “Copy result” copies the angle or the times found as text, and “Copy link” copies an address that contains your current input.

How the calculation works

How far each hand turns per minute

The minute hand makes one full turn (360°) in 60 minutes, so it moves 6° per minute. The hour hand makes one full turn in 12 hours, which is 720 minutes, so it moves 30° per hour and 0.5° per minute. Counting seconds, the minute hand moves 0.1° per second and the hour hand 1/120° per second (0.5° ÷ 60). The hands are treated as turning continuously at these speeds, not jumping from mark to mark.

Position of each hand

Positions are measured clockwise from the 12 o'clock direction, which is 0°. With h hours and m minutes the positions are as follows. The 12 starts from the same spot as 0, so for h use the remainder after dividing by 12.

• Hour hand = 30° × h + 0.5° × m (plus 0.5° × s/60 if there are seconds s)
• Minute hand = 6° × m (plus 0.1° × s if there are seconds s)

Smaller and larger angle

Call the difference between the two positions d. If d is 180° or less, d is the smaller angle. If d is over 180°, the smaller angle is 360° − d. There are two ways round the dial between the hands, and the two angles always add up to 360°. At 9:30 the hour hand is at 285° and the minute hand at 180°, a difference of 105°, so the smaller angle is 105° and the larger angle is 255°.

One-line formula

Subtracting the two positions and simplifying gives angle = |30h − 5.5m|. At 3:20 that is |90 − 110| = 20°, and at 12:30 it is |0 − 165| = 165°. If the result is over 180°, subtract it from 360°. At 10:10 it is |300 − 55| = 245°, so the smaller angle is 360° − 245° = 115°. The 5.5 is the minute hand's 6° per minute minus the hour hand's 0.5° per minute, that is, the angle the minute hand gains on the hour hand each minute.

Types of angle

The smaller angle runs from 0° to 180°. At 0° the hands overlap. An angle above 0° and below 90° is acute, 90° is a right angle, above 90° and below 180° is obtuse, and 180° is a straight angle with the hands in a line. An angle above 180° and below 360°, like the larger side here, is called a reflex angle.

Angle table by time

With a whole number of minutes the hour hand lands on a multiple of 0.5°, so values such as 7.5° or 5.5° are exact, not rounded. This is also why the hands do not overlap at 3:15. The minute hand is on the 3 (90°), but the hour hand has gone a further 7.5° in 15 minutes and is at 97.5°.

Finding when the hands overlap

A question such as “At what time between 5 and 6 do the hands overlap?” is a classic clock puzzle. At 5:00 the hour hand is at 150° and the minute hand at 0°. The minute hand gains 5.5° per minute on the hour hand, so the time x in minutes needed to make up 150° comes from 5.5x = 150, giving x = 300/11 = 27 3/11 minutes. The answer is 5:27 3/11, and since 3/11 of a minute is 16 4/11 seconds, that is about 5:27:16.4.

The same method gives the times between h o'clock and h+1 o'clock when the angle between the hands is θ. The minute hand must be θ short of, or θ past, the 30h° gap from the hour hand. In symbols 5.5x = 30h − θ or 5.5x = 30h + θ (add or subtract 360° if that passes a full turn), so x = 2(30h ± θ) ÷ 11 minutes. This is why most answers come out as a number of elevenths of a minute. The “Angle → time” mode of this calculator solves the equation in exact fractions and shows the mixed number and the seconds.

The hands overlap 11 times in 12 hours, evenly spaced 720/11 minutes apart, which is 65 5/11 minutes (about 1 hour 5 minutes 27 seconds). There is no overlap between 11 and 12 because the next one is exactly at 12:00. In a full day of 24 hours they overlap 22 times.

Straight lines also happen 11 times in 12 hours. Between 5 and 6 there is no moment when the hands are in a line, because that happens exactly at 6:00. A right angle (90°) normally occurs twice an hour, 22 times in 12 hours. Between 2 and 3 it happens only once, at 2:27 3/11, and the next one is exactly 3:00. Between 8 and 9 it is likewise 8:27 3/11, followed by 9:00.

So in one day the hands overlap 22 times, form a right angle 44 times and lie in a straight line 22 times. The minute hand turns 24 times a day and the hour hand twice, and the number of times the minute hand catches the hour hand is the difference, 22. You can see the same count in the graph at the bottom of the calculator by counting the points that touch 0° and 180°.

Common mistakes

• Fixing the hour hand on the number: answering 30° for 3:20, one gap between the 3 and the 4, is wrong. The hour hand also moved 10° in 20 minutes, so the answer is 20°. Giving 180° for 12:30 is the same mistake, and the correct answer is 165°.
• Giving the larger angle: when the difference in positions is over 180°, as at 10:10 (245°), subtract it from 360° to get the smaller angle (115°). Check first whether the question asks for the smaller angle between the hands.
• Mixing the angle between the hands with the angle a hand has turned: from 2:00 to 4:00 the minute hand turns 720°, yet at 4:00 the angle between the hands is 120°. How far a hand has moved and how far apart the hands are are different quantities.
• Giving overlap times in whole minutes: the overlap between 2 and 3 is at 2:10 10/11, not 2:11. At 2:11 the hour hand is at 65.5° and the minute hand at 66°, so they are 0.5° apart.
• Treating a.m. and p.m. or 24-hour times separately: 3:20 p.m. (15:20) and 3:20 a.m. put the hands in the same places, so the angle is the same. Midnight and noon are both counted as 12.

Limits

• The calculator assumes a 12-hour analog clock whose hands turn continuously at constant speeds. A real clock whose hands tick in steps can differ slightly from the calculated value depending on the moment you look.
• Times are entered as whole hours, minutes and seconds. Fractional seconds such as 0.5 are not accepted, but times found in “Angle → time” are shown exactly as fractions.
• The angle in “Angle → time” accepts up to three decimal places. Angles involving the second hand (hour and second, or minute and second) are not calculated.
• It cannot read a clock from a photo. Measuring a photo with a protractor can give a different value because of the thickness of the hands and the camera angle.
• The current time comes from this device's clock. Your input is calculated only in this browser and is not sent anywhere, though “Copy link” puts the input into the address.

Basis: the speeds of the hands and the definitions of the angle types do not change from year to year. The sources below were checked against the originals on 21 October 2026.

Sources

• 01University of Colorado Boulder, Math 1120 Review 3 SolutionsSolution 9: the hour hand moves 0.5° per minute and the minute hand 6° per minute (1/60 × 360°)
• 02NRICH (University of Cambridge), Watch the ClockThe hour hand moves (360 ÷ 12) ÷ 60 = 0.5° per minute, and the minute hand catching the hour hand is set up as an equation
• 03Wikipedia, Clock angle problemθ = |0.5° × (60H − 11M)|, subtracted from 360° if greater than 180°
• 04BIPM, The International System of Units (SI Brochure, 9th edition)Table 8: 1 h = 60 min = 3600 s, 1° = (π/180) rad
• 05Wolfram MathWorld, Obtuse Angle · Acute Angle · Straight AngleAn acute angle is less than 90°, an obtuse angle is more than 90° and less than 180°, and a straight angle is 180°
• 06University of Chicago, Everyday Mathematics: reflex angleReflex angle: an angle between 180° and 360°

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