Clock Angle Formula for SSC CGL (Quick Answer)
The angle between the hour hand and the minute hand at H hours M minutes is:
θ = |30H − 5.5M| (in degrees)
If the result is more than 180°, subtract it from 360° to get the smaller angle. Example: at 3:40, θ = |90 − 220| = 130°.
Key takeaway for SSC: The minute hand moves 6° per minute and the hour hand 0.5° per minute, so the minute hand gains 5.5° every minute. Every clock angle question, including "when do the hands coincide", is built on that one number. All answers on this page were computed with exact fractions in a program.
Clock questions appear in the reasoning sections of SSC CGL, SSC CHSL, SSC MTS, SSC GD and SSC CPO. For reflections of a clock, see our separate note on the mirror image of clock formula; for day-of-the-week questions, see the calendar odd days trick.
Speed of Clock Hands (Basic Facts)
| Fact | Value |
|---|---|
| Minute hand speed | 6° per minute (360° in 60 minutes) |
| Hour hand speed | 0.5° per minute (30° per hour) |
| Relative speed | 5.5° per minute (11/2 degree) |
| Angle between two hour marks | 30° |
| Minute spaces on the dial | 60 (each = 6°) |
| Minute hand gains on hour hand | 55 minute spaces every hour |
Angle Between Hands of a Clock: Solved Examples
| Time | 30H − 5.5M | Angle (smaller) | Reflex angle |
|---|---|---|---|
| 3:40 | 90 − 220 = −130 | 130° | 230° |
| 7:20 | 210 − 110 = 100 | 100° | 260° |
| 9:15 | 270 − 82.5 = 187.5 | 172.5° | 187.5° |
| 4:30 | 120 − 165 = −45 | 45° | 315° |
| 2:45 | 60 − 247.5 = −187.5 | 172.5° | 187.5° |
| 10:10 | 300 − 55 = 245 | 115° | 245° |
| 11:50 | 330 − 275 = 55 | 55° | 305° |
| 5:24 | 150 − 132 = 18 | 18° | 342° |
| 8:18 | 240 − 99 = 141 | 141° | 219° |
Watch out for 9:15 and 2:45: the raw value is above 180°, so the smaller angle is 360° − 187.5° = 172.5°, not 187.5°.
When Do the Hands Coincide, Become Opposite or Make a Right Angle?
Between H and (H + 1) o'clock, the minute hand is at these minutes past H:
| Position of hands | Minutes past H |
|---|---|
| Coincide (0°) | 60H/11 |
| Opposite (180°) | (5H ± 30) × 12/11 |
| Right angle (90°) | (5H ± 15) × 12/11 |
| Any angle θ | (60H ± 2θ)/11 |
Keep only the answers between 0 and 60 minutes.
Example 1: Hands together between 4 and 5
Minutes past 4 = 60 × 4/11 = 240/11 = 21 9/11 minutes past 4.
Example 2: Hands opposite between 4 and 5
(5 × 4 + 30) × 12/11 = 600/11 = 54 6/11 minutes past 4. (The "− 30" value is negative, so it is rejected.)
Example 3: Right angle between 7 and 8
(35 − 15) × 12/11 = 240/11 = 21 9/11 minutes past 7, and (35 + 15) × 12/11 = 600/11 = 54 6/11 minutes past 7. Both are valid, so the hands are at right angles twice.
Example 4: Right angle between 2 and 3
(10 + 15) × 12/11 = 300/11 = 27 3/11 minutes past 2. The other value is negative, so there is only one right angle strictly between 2 and 3. The same happens between 8 and 9, because the right angles at exactly 3:00 and 9:00 belong to the next hour.
How Many Times Do the Hands Coincide in a Day?
| Event | In 12 hours | In 24 hours |
|---|---|---|
| Hands coincide | 11 | 22 |
| Hands opposite (straight line, 180°) | 11 | 22 |
| Hands at right angles | 22 | 44 |
| Hands in a straight line (0° or 180°) | 22 | 44 |
The hands coincide every 720/11 = 65 5/11 minutes, not every 60 minutes, which is why there are 11 and not 12 meetings in 12 hours.
Faulty Clock Questions (Gain and Loss)
Example 5. The hands of a clock coincide every 64 minutes of correct time. How much does the clock gain or lose in a day?
- In a correct clock, the hands coincide every 65 5/11 minutes.
- Here they coincide every 64 true minutes, so the clock runs fast.
- In 24 hours (1440 true minutes), the clock shows 1440 × (720/11) ÷ 64 = 1472 8/11 minutes.
- The clock gains 32 8/11 minutes in a day.
Common Mistakes in Clock Angle Questions
- Using 6M instead of 5.5M: the hour hand also moves while the minute hand moves.
- Reporting the reflex angle (above 180°) when the question asks for "the angle between the hands".
- Taking 60 minutes as the gap between two meetings of the hands (it is 65 5/11 minutes).
- Forgetting the second answer in right-angle and angle-θ questions.
- Using H = 12 incorrectly: at 12:36 use H = 12 or H = 0; both give 162°.
Quick Revision Table: Clock Angle Tricks
| Question | Formula |
|---|---|
| Angle at H:M | 30H − 5.5M without sign, then 360° minus it if above 180° |
| Hands together | 60H/11 minutes past H |
| Hands opposite | (5H ± 30) × 12/11 minutes past H |
| Right angle | (5H ± 15) × 12/11 minutes past H |
| Angle θ | (60H ± 2θ)/11 minutes past H |
| Gap between meetings | 720/11 = 65 5/11 minutes |
| Meetings per day | 22 coincide, 22 opposite, 44 right angles |
Clock Angle Practice Questions with Answers
- Find the angle between the hands at 6:35. Answer: 12.5°. |180 − 192.5| = 12.5°.
- Find the angle between the hands at 1:55. Answer: 87.5°. |30 − 302.5| = 272.5°, so the smaller angle is 360° − 272.5° = 87.5°.
- Find the angle between the hands at 12:36. Answer: 162°. |0 − 198| = 198°, so 360° − 198° = 162°.
- At what time between 5 and 6 are the hands together? Answer: 27 3/11 minutes past 5. 60 × 5/11 = 300/11.
- At what time between 8 and 9 are the hands opposite each other? Answer: 10 10/11 minutes past 8. (40 − 30) × 12/11 = 120/11.
- At what times between 6 and 7 is the angle between the hands 60°? Answer: 21 9/11 and 43 7/11 minutes past 6. (360 ± 120)/11 = 240/11 and 480/11.
- How many times are the hands at right angles in a day? Answer: 44.
For timed practice, take the SSC CGL Calendar & Clock Mock Test 02 and the SSC CHSL Calendar & Clock Mock Test 01.
Clock Angle FAQs
What is the formula for the angle between the hands of a clock?
θ = |30H − 5.5M| degrees, where H is the hour and M the minutes. If θ is more than 180°, the smaller angle is 360° − θ.
How many times do the hands of a clock coincide in 24 hours?
22 times. They meet every 65 5/11 minutes, so only 11 times in 12 hours.
How many times are the hands of a clock at right angles in a day?
44 times in 24 hours (22 in 12 hours).
Why does the minute hand gain 5.5 degrees per minute?
The minute hand moves 6° per minute and the hour hand 0.5° per minute, so the gap changes by 6 − 0.5 = 5.5° every minute.
Practice Next
- Topic tests: SSC CGL Calendar & Clock Mock Test 01, SSC CGL Calendar & Clock Mock Test 02, SSC CHSL Calendar & Clock Mock Test 01 and SSC GD Calendar and Clock Mock Test 01.
- Related notes: mirror image of clock formula, calendar odd days trick and mathematical operations questions for SSC CGL.
- Browse all study notes.