Finding the Clock Angle at 8:20

This problem requires us to determine the angle formed between the hour hand and the minute hand of a clock when the time is 8:20. We need to understand how each hand moves on the clock face.

Understanding Clock Hand Movement

A clock face is a circle of 360 degrees. It is divided into 12 hours, meaning the angle between two consecutive hour marks is $360^{\circ} / 12 = 30^{\circ}$. It is also divided into 60 minutes.

• Minute Hand Movement: The minute hand completes a full circle (360 degrees) in 60 minutes. Therefore, its speed is $360^{\circ} / 60 \text{ minutes} = 6^{\circ}$ per minute.
• Hour Hand Movement: The hour hand completes a full circle (360 degrees) in 12 hours (720 minutes). Therefore, its speed is $360^{\circ} / 12 \text{ hours} = 30^{\circ}$ per hour. This is equivalent to $30^{\circ} / 60 \text{ minutes} = 0.5^{\circ}$ per minute.

Calculating Hand Positions at 8:20

To find the angle, we calculate the position of each hand relative to the 12 o'clock position (0 degrees).

Minute Hand Position

At 8:20, the minute hand points directly at the 20-minute mark.

Position of Minute Hand = Number of minutes $\times$ Degrees per minute

Position of Minute Hand = $20 \times 6^{\circ} = 120^{\circ}$

Hour Hand Position

At 8:20, the hour is 8 and the minutes are 20.

The hour hand's position depends on both the hour and the minutes past the hour.

Position of Hour Hand = (Hour $\times$ Degrees per hour) + (Minutes $\times$ Degrees per minute for hour hand)

Position of Hour Hand = $(8 \times 30^{\circ}) + (20 \times 0.5^{\circ})$

Position of Hour Hand = $240^{\circ} + 10^{\circ} = 250^{\circ}$

Calculating the Angle Between Hands

The angle between the hands is the absolute difference between their positions.

Angle = |Position of Hour Hand - Position of Minute Hand|

Angle = $|250^{\circ} - 120^{\circ}|$

Angle = $130^{\circ}$

Since the angle $130^{\circ}$ is less than $180^{\circ}$, this is the required angle.

Formulaic Approach

A common formula to calculate the angle between the hour hand (H) and minute hand (M) is:

Angle = $\frac{1}{2} |60H - 11M|$

Here, H = 8 and M = 20.

Angle = $\frac{1}{2} |60 \times 8 - 11 \times 20|$

Angle = $\frac{1}{2} |480 - 220|$

Angle = $\frac{1}{2} |260|$

Angle = $130^{\circ}$

Both methods yield the same result.