Calculating the Angle Between Clock Hands at 8 PM

This problem requires us to find the angle formed by the hour hand and the minute hand of a clock when the time is exactly 8:00 p.m.

To solve this, we need to understand how the hour hand and the minute hand move relative to the clock face.

Understanding Clock Hand Movement

• The minute hand completes a full circle (360 degrees) in 60 minutes.
• The hour hand completes a full circle (360 degrees) in 12 hours.

Calculating Speed of Hands

• Minute Hand Speed:
  • In 60 minutes, it moves $360^\circ$.
  • In 1 minute, it moves $\frac{360^\circ}{60} = 6^\circ$.
• Hour Hand Speed:
  • In 12 hours, it moves $360^\circ$.
  • In 1 hour, it moves $\frac{360^\circ}{12} = 30^\circ$.
  • In 1 minute, it moves $\frac{30^\circ}{60} = 0.5^\circ$.

Determining Position at 8:00 PM

At 8:00 p.m., the time is exactly on the hour. This simplifies the calculation.

• The minute hand is pointing directly at the 12. We can consider this position as $0^\circ$ (or $360^\circ$).
• The hour hand is pointing directly at the 8.

Calculating Angle of Each Hand from the 12

We calculate the angle of each hand clockwise from the 12 o'clock position.

• Minute Hand Angle:
  • At 8:00, the minute hand is at 12.
  • Angle of minute hand $= 0^\circ$.
• Hour Hand Angle:
  • The clock face is divided into 12 hours, with $360^\circ$ in total.
  • The angle between consecutive hour marks (like 12 and 1, or 1 and 2) is $\frac{360^\circ}{12} = 30^\circ$.
  • At 8:00, the hour hand is pointing exactly at the 8.
  • The angle from the 12 to the 8 is $8$ hour marks.
  • Angle of hour hand $= 8 \times 30^\circ = 240^\circ$.

Finding the Angle Between the Hands

The angle between the two hands is the absolute difference between their angles from the 12.

Angle $= | \text{Angle of Hour Hand} - \text{Angle of Minute Hand} |$

Angle $= | 240^\circ - 0^\circ | = 240^\circ$

Clocks have two angles between the hands: the smaller angle (usually $\le 180^\circ$) and the larger reflex angle (usually $\ge 180^\circ$). When asked for "the angle", it typically refers to the smaller one.

The smaller angle is $360^\circ - \text{Larger Angle}$.

Smaller Angle $= 360^\circ - 240^\circ = 120^\circ$

Comparing this with the given options, $120^\circ$ is one of the choices.

The difference in angles is $|240^\circ - 0^\circ| = 240^\circ$. The smaller angle is $360^\circ - 240^\circ = 120^\circ$.

Conclusion

The angle between the two hands of a clock when the time shown is 8 p.m. is $120^\circ$.

Revision Table: Clock Angle at 8 PM

Additional Information: General Clock Angle Formula

For any time H hours and M minutes, the angle between the hands can be calculated using a general formula.

• Angle of Minute Hand from 12: $6 \times M$ degrees
• Angle of Hour Hand from 12: $30 \times H + 0.5 \times M$ degrees

The angle between the hands is $| (30 \times H + 0.5 \times M) - (6 \times M) |$ degrees.

This simplifies to $| 30 \times H - 5.5 \times M |$ degrees.

For 8:00 p.m., H=8 and M=0.

Angle $= | 30 \times 8 - 5.5 \times 0 |$

Angle $= | 240 - 0 | = 240^\circ$

The smaller angle is $360^\circ - 240^\circ = 120^\circ$. This confirms our previous step-by-step calculation for the angle at 8:00 p.m.

Remember that the formula gives the absolute difference in angles, and you might need to subtract from $360^\circ$ to find the smaller angle if the result is greater than $180^\circ$.