Introduction

To understand the angle between the hour and minute hands on a clock, we can use a straightforward mathematical approach. Clock angles are a classic problem in analytical geometry and help in understanding time-telling intricacies. This guide will walk you through the steps to calculate the angle at 3:15, provide some additional examples, and explain the underlying principles.

Calculating the Angle Between the Hour and Minute Hands

The angle between the hour and minute hands of a clock can be calculated using the following steps:

Minute Hand Movement

The minute hand of a clock moves 360 degrees in 60 minutes. Therefore, the position of the minute hand at any given time can be calculated as follows:

Minute hand angle  (minute / 60) * 360

Minute hand angle  (15 / 60) * 360  90 degrees

Hour Hand Movement

The hour hand moves 360 degrees in 12 hours, which is 30 degrees per hour. Additionally, the hour hand moves continuously as the minutes pass, moving 0.5 degrees per minute. Therefore, the position of the hour hand can be calculated as follows:

Hour hand position at 3:00  3 * 30  90 degreesAdditional movement for 15 minutes: 15 * 0.5  7.5 degreesTotal hour hand position at 3:15  90   7.5  97.5 degrees

Angle Between Two Hands

The angle between the hour and minute hands is the absolute difference between their positions:

Angle  |Hour hand position - Minute hand position|

Angle  |97.5 - 90|  7.5 degrees

Additional Examples

Example 1: 3:20
At 3:20, the minute hand will be at 4 (since 20 minutes is one-third of an hour, the minute hand will be at the 4 o'clock position).

The hour hand will have moved 20 minutes from the 3 o'clock position. Since the hour hand moves 0.5 degrees per minute, it will have moved:

20 * 0.5  10 degrees

The angle between the 3 o'clock and 4 o'clock positions is 30 degrees. However, the hour hand has moved 10 degrees towards the 4 o'clock position, reducing the angle:

Angle between the hands  30 - 10  20 degrees

Example 2: 3:30
At 3:30, the minute hand is exactly at the 6 o'clock position. The hour hand will have moved halfway to the 4 o'clock position:

30 * 0.5  15 degrees

The angle between the hands is:

Angle between the hands  30 - 15  15 degrees

Conclusion

Understanding the movement of the clock hands is a fundamental concept in competitive programming and mathematics. By breaking down the problem into smaller components and applying the appropriate formulas, you can quickly and accurately calculate the angle between the hour and minute hands at any given time.

This guide has provided you with the necessary steps to calculate the angle for the example of 3:15 and additional examples. With practice, you will be able to apply these principles to a variety of clock angle problems.