Understanding the Angle Between Clock Hands at 8:10

Determining the angle between the hour and minute hands of a clock is a common problem that requires a combination of basic geometry and time understanding. This guide will walk you through the precise steps to calculate the angle at 8:10. We will use both a detailed step-by-step approach and a more concise method to arrive at the correct answer. Moreover, we will provide a clear and concise explanation of the mathematical reasoning behind the calculations.

Step-by-Step Calculation

Let's break down the process of finding the angle at 8:10 using a detailed approach:

1. Position of the Hour Hand

Each hour represents 30 degrees on a clock because there are 360 degrees in a circle and 12 hours. Thus, we have:

Each hour 360 degrees / 12 hours 30 degrees per hour

At 8:00, the hour hand is at 8 times 30 degrees 240 degrees.

As time progresses, the hour hand moves. In 10 minutes, the hour hand moves an additional 10 times 0.5 degrees 5 degrees since it moves 0.5 degrees per minute.

Therefore, at 8:10, the hour hand is at 240 5 245 degrees.

2. Position of the Minute Hand

Each minute represents 6 degrees since there are 360 degrees in a circle and 60 minutes. Thus, we have:

Each minute 360 degrees / 60 minutes 6 degrees per minute

At 8:10, the minute hand is at 10 times 6 degrees 60 degrees.

3. Calculating the Angle

The angle between the two hands can be found by taking the absolute difference between their positions:

Angle 245 - 60 185 degrees

4. Determining the Smaller Angle

Given that the maximum angle between the two hands can be 360 degrees, we need to check if the calculated angle is greater than 180 degrees. If it is, we subtract it from 360 degrees to find the smaller angle:

Smaller Angle 360 - 185 175 degrees

Minute-by-Minute Calculation

Another way to calculate the angle is by considering the total minute markings and the movement of the hands:

At 8:00, the small hand is at 240 degrees (number 8) and the big hand is at 360 degrees (number 12). The angle between them is 240 degrees.

When 8:10 strikes, the minute hand moves by 5 degrees (10 minutes * 0.5 degrees per minute).

Thus, the angle between them is 240 - 5 175 degrees.

Mathematical Formulation

Assume the time as h hours and m minutes. Here, h 8 and m 10.

The minute hand rotates 6 degrees for 1 minute:

Minute hand angle 6 m degrees

The hour hand rotates 30 degrees for 1 hour and 0.5 degrees for 1 minute:

Hour hand angle 30 h 0.5 m degrees

Thus, the angle between the two hands is:

Angle 30 h 0.5 m - 6 m 245 - 60 185 degrees

Given that 185 degrees is greater than 180 degrees, the smaller angle is:

Smaller Angle 360 - 185 175 degrees

Therefore, the angle at 8:10 is 175 degrees.