clock angle problem

What is the measure of the small angle formed between the hour hand and the minute hand? note: The angle is measured in degrees. The time is based on a 12-hour clock.

solution

• The clock is circular in shape, this means one round is 360°
• The hour hand rotates through 360° in 12 hours or 30° per hour (360 / 12 = 30)
• The minute hand rotates through 360° in 60 minutes or 6° per minute (360 / 60 = 6)
• When the minute hand rotates the hour hand will also rotate
• The angle between the hands = | angle formed by hour hand - angle formed by the minute hand | (we take the absolute value to guarantee a positive angle)
• to get the measure of the small angle then subtract it from 360 degrees if it is greater than 180 degrees.

example

1. What is the angle between the hour hand and the minute hand at 3:15?

• We have 3 hours and 15 minutes, converting the 15 minutes into hours gives us 0.25 hours, so we have 3.25 hours
• The angle formed by the hour hand = 3.25 hours * 30° = 97.5°
• We have 15 minutes, so
• The angle formed by the minute hand = 15 minute * 6° = 90°
• The angle formed between the two hands = |97.5° - 90°| = 7.5°

2. What is the angle between the hour hand and the minute hand at 1:45?

• We have 1 hour and 45 minutes, converting the 45 minutes into hours gives us 0.75 hours, so we have 1.75 hours
• The angle formed by the hour hand = 1.75 hours * 30° = 52.5°
• We have 45 minutes, so
• The angle formed by the minute hand = 45 minute * 6° = 270°
• The angle formed between the two hands = |52.5° -270°| = 217.5°
• The angle is over 180°, so to get the smaller angle simply, 360° - 217.5° = 142.5°

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