Clock-angle puzzles are classic interview and exam questions. The trick is that the hour hand moves too: at 3:15 it has gone a quarter of the way to 4, so the hands are 7.5° apart, not 0°.
How to use the Clock Angle Calculator
- Hours
- Minutes
- Seconds
Results update as you type, and everything is calculated in your browser.
How it works
Enter hours (0–23), minutes and optionally seconds. Each hand’s position is measured clockwise from 12; the difference gives the angle in each direction.
Tips
- Angles are measured counter-clockwise from the positive x-axis by default.
- Add or subtract 360° (2π) to move between coterminal angles.
- Complementary angles add to 90°, supplementary angles to 180°.
- Use the unit menu to switch between degrees, radians and gradians.
Common mistakes to avoid
- Mixing up complementary and supplementary angles.
- Forgetting that a reference angle is always between 0° and 90°.
- Leaving an angle in radians when the question asks for degrees.
Inputs explained
| Field | What to enter |
|---|---|
| Hours | A number |
| Minutes | A number |
| Seconds | A number |
What each result means
- Hour hand position = 30×mod([Hours], 12) + 0.5×[Minutes] + [Seconds]÷120
- Minute hand position = 6×[Minutes] + 0.1×[Seconds]
- Angle from the hour hand to the minute hand (clockwise) = mod([Minute hand position] – [Hour hand position], 360)
- Angle from the minute hand to the hour hand (clockwise) = mod([Hour hand position] – [Minute hand position], 360)
- Smaller angle between the hands = min([Angle from the hour hand to the minute hand (clockwise)], [Angle from the minute hand to the hour hand (clockwise)])
- Reflex angle = 360 – [Smaller angle between the hands]
Example values
How the main result changes when Hours changes and everything else stays at the example values above:
| Hours | Smaller angle between the hands (°) |
|---|---|
| 1.5 | 37.5 |
| 2.25 | 15 |
| 3 | 7.5 |
| 3.75 | 30 |
| 4.5 | 52.5 |
| 6 | 97.5 |
Formula
Hour hand: 30°·h + 0.5°·m + s/120 · Minute hand: 6°·m + 0.1°·s
Angle = |hour − minute|, smaller angle = min(angle, 360° − angle)
Worked example
Inputs: Hours: 3; Minutes: 15; Seconds: 0.
Results: Hour hand position = 97.5 °; Minute hand position = 90 °; Angle from the hour hand to the minute hand (clockwise) = 352.5 °; Angle from the minute hand to the hour hand (clockwise) = 7.5 °; Smaller angle between the hands = 7.5 °; Reflex angle = 352.5 °.
Key terms
- Reflex angle
- An angle between 180° and 360°.
Frequently asked questions
What is the angle at 3:15?
7.5°. The hour hand has moved past 3.
How often do the hands overlap?
11 times in 12 hours, every 65 5/11 minutes.
When are the hands at right angles?
22 times in 12 hours.
How fast does each hand move?
Minute hand 6° per minute, hour hand 0.5° per minute.