Clock Angle Calculator
Calculate angles between hour and minute hands on analog clocks with step-by-step solutions
Calculate Clock Angles
Enter hours from 1 to 12
Enter minutes from 0 to 59
Current Time
Clock Angles
Hour Hand Position
360°
From 12 o'clock position
Minute Hand Position
0°
From 12 o'clock position
Angle Classification
Example Problems
Example 1: Simple Case
Problem: What is the angle between clock hands at 3:00?
Solution:
• Minute hand: 6° × 0 = 0°
• Hour hand: 30° × 3 + 0.5° × 0 = 90°
• Angle difference: |90° - 0°| = 90°
Answer: 90° (right angle)
Example 2: Complex Case
Problem: Find the angle at 10:14.
Solution:
• Minute hand: 6° × 14 = 84°
• Hour hand: 30° × 10 + 0.5° × 14 = 300° + 7° = 307°
• Angle difference: |307° - 84°| = 223°
• Since 223° > 180°, smaller angle = 360° - 223° = 137°
Answer: 137° and 223°
Clock Angle Formulas
Large: max(α, 360° - α)
Clock Facts
Hour hand moves 0.5° per minute
Minute hand moves 6° per minute
Hands overlap 11 times in 12 hours
Right angle at 3:00 and 9:00
Straight line at 6:00
Quick Reference
Used in geometry and trigonometry
Common in math competitions
Practical application of angles
Time and motion concepts
Understanding Clock Angles
How Clock Hands Move
Understanding clock angles starts with knowing how fast each hand moves. The minute hand completes a full 360° rotation in 60 minutes, moving 6° per minute. The hour hand completes a full rotation in 12 hours (720 minutes), moving 0.5° per minute.
Two Methods to Solve
You can solve clock angle problems using logical reasoning or mathematical formulas. The logical method involves visualizing the clock and counting degrees, while the formula method uses precise calculations.
Why Two Angles?
- •Clock hands divide the circle into two arcs
- •Both angles always sum to 360°
- •The smaller angle is usually the answer requested
- •Both angles are geometrically significant
Mathematical Foundation
Degree Distribution:
• Full circle: 360°
• Each hour: 360° ÷ 12 = 30°
• Each minute: 360° ÷ 60 = 6°
• Hour hand per minute: 30° ÷ 60 = 0.5°
Special Times
12:00: 0° (hands overlap)
3:00: 90° (right angle)
6:00: 180° (straight line)
9:00: 90° (right angle)
1:05: 0° (hands overlap again)
Educational Value
Clock angle problems teach students about circular motion, angular velocity, and the relationship between time and angles.
Problem Solving
These problems develop spatial reasoning, mathematical visualization, and the ability to break complex motions into simple components.
Real Applications
Clock angles appear in engineering, astronomy, navigation, and any field involving rotational motion and timing.