Calculating Clock Hand Angle at 5:36 AM

Understanding how to calculate the angle between the hour hand and the minute hand of a clock involves knowing their relative speeds and positions at a given time. Let's break down the process for 5:36 AM.

Clock Hand Speeds

• The minute hand completes a full circle (360°) in 60 minutes. Its speed is $\frac{360}{60} = 6$ degrees per minute.
• The hour hand completes a full circle (360°) in 12 hours. Its speed is $\frac{360}{12} = 30$ degrees per hour. Converting this to degrees per minute: $\frac{30}{60} = 0.5$ degrees per minute.

Position of the Minute Hand at 5:36 AM

At 36 minutes past the hour, the minute hand's position is calculated based on its speed from the 12 o'clock mark.

Minute hand position = Minutes $\times$ Speed of minute hand

Minute hand position = $36 \times 6^{\circ}/\text{minute}$

Minute hand position = $216^{\circ}$ from the 12.

Position of the Hour Hand at 5:36 AM

The hour hand's position depends on both the hour and the minutes past the hour. At 5:00, the hour hand is exactly at the 5. Each hour mark is $30^{\circ}$ apart ($360^{\circ}/12$). The hour hand moves an additional amount based on the minutes.

• Angle due to the hour = Hour $\times 30^{\circ}$
• Angle due to the minutes = Minutes $\times 0.5^{\circ}/\text{minute}$

At 5:36 AM:

Hour hand position = $(5 \times 30^{\circ}) + (36 \times 0.5^{\circ})$

Hour hand position = $150^{\circ} + 18^{\circ}$

Hour hand position = $168^{\circ}$ from the 12.

Calculating the Angle Between the Hands

The angle between the hour hand and the minute hand is the absolute difference between their positions.

Angle = $\mid \text{Hour hand position} - \text{Minute hand position} \mid$

Angle = $\mid 168^{\circ} - 216^{\circ} \mid$

Angle = $\mid -48^{\circ} \mid$

Angle = $48^{\circ}$

The angle between the hands is $48^{\circ}$. Since $48^{\circ}$ is less than $180^{\circ}$, this is the smaller angle. If the result were greater than $180^{\circ}$, we would subtract it from $360^{\circ}$ to find the smaller angle.

Let's summarize the positions:

The difference in angles is $|216^{\circ} - 168^{\circ}| = 48^{\circ}$.

Revision Table: Key Concepts for Clock Angles

Remember these speeds are crucial for solving clock angle problems for any given time.

Additional Information: General Formula for Clock Angles

You can use a general formula to find the angle between clock hands at H hours and M minutes:

Angle = $\mid (30 \times \text{H}) - (11/2 \times \text{M}) \mid$

Or, expressed differently:

Angle = $\mid (30 \times \text{H}) - (5.5 \times \text{M}) \mid$

Let's check this for 5:36 AM (H=5, M=36):

Angle = $\mid (30 \times 5) - (5.5 \times 36) \mid$

Angle = $\mid 150 - 198 \mid$

Angle = $\mid -48 \mid$

Angle = $48^{\circ}$

This formula gives the same result and is a quick way to calculate the angle between the hour hand and the minute hand.