Find the angle between the hour hand and the minute hand of a clock at 8:20.
130°
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.
Similar Questions
If 18 th February, 2005 falls on Friday then what will be the day on 18th February, 2007?
It was Sunday on Jan. 1, 2006. What was the day of the week on Jan.1, 2010?
- A clock is set right at 8 a.m. The clock gains 10 minutes in 24 hours. What will be the true time when the clock indicates 1 p.m. on the following day?
- The reflex angle between the hands of a clock at $10:25$ is
If the 25 th of August in a year is Thursday, the number of Mondays in that month is
Find the angle between the hour hand and the minute hand of a clock at 6 ∶ 10
Which of the following is the odd one out ?
If October 10, 2023 was a Tuesday, then what day of the week was December 23, 2023 ?
If 1st January, 2024 was a Monday, then which day of the week was 1st January, 2025 ?
How many times in a day do the hands of a clock form a straight line but point in opposite directions ?
Important Questions from Clock and Calendar
- The calendar for the year 1733 was the same for the year:
- Y's 7th birthday is on 10th May which falls on Thursday of a leap year. What will be the day on his 10th birthday?
- If 8th June, 2037 is Sunday, what was the day of the week on 8th June, 2036?
- Today is Tuesday. Before 28 days, it was:
- If the first day of 1812 was a Wednesday, what day was the last day of that year?
