🕐 Clock Angle Calculator
Calculate the angle between the hour and minute hands
How to Use This Calculator
Enter Hours
Input the hour (0-23 for 24-hour format, or 1-12 for 12-hour format). The calculator accepts both formats.
Enter Minutes
Input the minutes (0-59). This determines how far the minute hand has moved.
Calculate
Click "Calculate Clock Angle" to find the angle between the hour and minute hands in degrees and radians.
Formula
Angle = |(Hour × 30° + Minutes × 0.5°) - (Minutes × 6°)|
The result is the smaller angle (always ≤ 180°)
Hour Hand Movement:
- Hour hand moves 30° per hour (360° / 12 hours = 30°/hour)
- Hour hand also moves 0.5° per minute (30° / 60 minutes = 0.5°/minute)
- Hour Angle = (Hour × 30°) + (Minutes × 0.5°)
Minute Hand Movement:
- Minute hand moves 6° per minute (360° / 60 minutes = 6°/minute)
- Minute Angle = Minutes × 6°
Example 1: 3:00 (3 hours, 0 minutes)
Hour Angle = 3 × 30° = 90°
Minute Angle = 0 × 6° = 0°
Angle = |90° - 0°| = 90°
Example 2: 3:30 (3 hours, 30 minutes)
Hour Angle = (3 × 30°) + (30 × 0.5°) = 90° + 15° = 105°
Minute Angle = 30 × 6° = 180°
Angle = |105° - 180°| = 75°
(Since 75° < 180°, this is the smaller angle)
Example 3: 12:15
Hour Angle = (12 × 30°) + (15 × 0.5°) = 360° + 7.5° = 367.5° (mod 360 = 7.5°)
Minute Angle = 15 × 6° = 90°
Angle = |7.5° - 90°| = 82.5°
About Clock Angle Calculator
The Clock Angle Calculator finds the angle between the hour and minute hands of an analog clock at any given time. This is a classic geometry problem that demonstrates how time relates to angular positions.
When to Use This Calculator
- Geometry Problems: Solve clock angle puzzles and problems
- Education: Teach students about angles and time relationships
- Puzzles: Solve brain teasers involving clock angles
- Interviews: Practice for technical interviews with clock problems
- Design: Create clock displays and animations
- Physics: Understand angular velocity and position
Why Use Our Calculator?
- ✅ Quick Results: Calculate angle instantly
- ✅ 24/12 Hour Format: Supports both time formats
- ✅ Detailed Breakdown: Shows hour and minute hand positions
- ✅ Degrees & Radians: Results in both units
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Clock Angles
On an analog clock, both hands move continuously:
- Hour Hand: Completes 360° in 12 hours (30°/hour). Also moves 0.5° per minute as the hour progresses.
- Minute Hand: Completes 360° in 60 minutes (6°/minute).
- Angle Calculation: The angle between hands is the absolute difference between their positions, always taking the smaller angle (≤ 180°).
- Special Times: At 12:00, both hands overlap (0°). At 6:00, they form a straight line (180°).
Real-World Applications
Clock Design: Designers use clock angle calculations to ensure proper hand positioning and create accurate time displays.
Animation: Animators calculate clock angles to create realistic clock hand movements in digital clocks and games.
Puzzles: Clock angle problems are popular in math competitions and brain teasers, testing understanding of angles and time.
Frequently Asked Questions
How do you calculate the angle between clock hands?
Calculate each hand's position: Hour Angle = (Hour × 30°) + (Minutes × 0.5°), Minute Angle = Minutes × 6°. Then find the absolute difference and take the smaller angle (≤ 180°).
Why does the hour hand move during minutes?
The hour hand moves continuously, not just at the hour. As minutes pass, the hour hand moves from its current hour position toward the next hour. This movement is 0.5° per minute (30°/hour ÷ 60 minutes).
What time shows 90 degrees between hands?
Many times show 90°! Examples: 3:00, 9:00, 3:16.36, 9:49.09. The exact times depend on how precisely the hour hand has moved past the hour mark.
When are the clock hands at 180 degrees?
The hands form a straight line (180°) at 6:00 exactly, and at other times like 12:32.73, 1:38.18, etc. These are times when the hands are directly opposite each other.
Can the angle be greater than 180°?
The calculator always returns the smaller angle (≤ 180°). If the difference is greater than 180°, it subtracts from 360° to get the smaller angle.
How many times do the hands overlap in 12 hours?
The hour and minute hands overlap 11 times in 12 hours (not 12, because they overlap just before 12:00, not exactly at 12:00). The overlaps occur approximately every 65.45 minutes.
What's the formula for the hour hand position?
Hour hand angle = (Hour × 30°) + (Minutes × 0.5°). The hour is taken modulo 12 (so 12 becomes 0, 13 becomes 1, etc.).
How fast does each hand move?
The hour hand moves 30° per hour (0.5° per minute). The minute hand moves 6° per minute (360° per hour). The minute hand moves 12 times faster than the hour hand.