What are the Different Clocks Angles?
Two distinct measurements apply to clocks angle problems: angles and time. Usually, the clocks angles are determined in degrees from the clockwise mark of number 12. Usually, the time is based on a 12-hour clock.
The rate of change of the angle in degrees per minute is a technique to solve such problems. A typical 12-hour analog clock's hour hand turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates 60 minutes or 6° per minute through 360°.
What is a Clock?
A clock is a system that is used to calculate, preserve, and show time. One of the oldest human inventions is the clock, which satisfies the need to calculate time intervals shorter than natural units: the day, the lunar month, and the year.
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What is Hour Hand?
A tiny hand on a clock showing the hours. It goes around the clock once every 12 hours (half a day). Example: In the below image, the hour hand is at 6 in the lower clock, so the time is 6 o'clock.
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The clock angle formula for the hour-hand angle:
θhr = 0.5° × MΣ
θhr = 0.5° × (60 × H + M)
where:
The angle of the hand measured clockwise from the 12 as the angle in degrees is θ.
The hour is taken as H.
M is the minutes of the hour after the hour.
MΣ is the number of minutes from 12 o'clock onwards.
What is Minute Hand?
A big hand on a timepiece that points to the minutes.
It goes around the clock once every 60 minutes (one hour).
For eg, the minute hand is just past the "7" in the clock below, and if you count the small marks of "12" it shows that it's 37 minutes past the hour.
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Basic Concept of Clock Formula to Find Angle
A clock is a complete 360-degree circle. It is split into 12 equal parts, i.e. 360/12 = 30° per part.
As the minute hand takes a full round in one hour, in 60 minutes, it covers 360°.
360/60 = 6°/ minute covers it in 1 minute.
Also, as the hand of the hour covers only one part in one hour out of the provided 12 parts. This means that 30° is covered in 60 minutes, i.e. 1⁄2° per minute.
How to Calculate the Angle Between Hour and Minute Hand?
The angle between hour and minute hand formula can be calculated as:
The idea is to take as a reference 12:00 (h = 12, m = 0). Detailed steps to find the angle between hour hand and minute hand are provided below.
Calculate the angle in hours and minutes made by hour hand with respect to 12:00.
Calculate the angle made by the minute hand in hours and minutes with respect to 12:00.
The difference between the two angles is the angle between the minute hand and the hour hand.
How can the Two Angles be Calculated with Respect to 12:00?
The minute hand moves in 60 minutes a measure of 360 degrees (or 6 degrees in one minute) and the hour hand in 12 hours moves 360 degrees (or 0.5 degrees in 1 minute). The minute hand would move (h*60 + m)*6 within hours and m minutes and the hour hand would move (h*60 + m)*0.5.
Let us look at some clock angle problems with solutions to understand the topic better.
Solved Examples
1. What measure is the angle between minute hand and hour hand at 4 o'clock?
Solution:
The minute hand is on the 12th and the hour hand is on the 4th at four o'clock. In a circle, the angle shaped is 4/12 of the total number of degrees, 360.
4/12 * 360 = 120 degrees.
2. What angle do the hands render if a clock reads 8:15 PM?
Solution:
A clock is a circle, and 360 degrees is still a circle. Since a clock lasts 60 minutes, each minute mark is 6 degrees.
\[\frac{360^{\circ}}{60 \text{ minutes}} = 6 \text{ degrees per minute}\]
The minute hand will point to 15 minutes on the clock, allowing us to determine the circle's location.
(15 min) × 6 = 90°
Each hour mark is 30 degrees since there are 12 hours on the clock,
\[\frac{360^{\circ} total}{12} = 30 \text{ degrees per hour}\]
We can calculate where the hand is going to be at 8:00 a.m.
(8 hr) × 30 = 240°
However, when we look at 8:15 rather than an absolute hour mark, the hour hand would probably be between 8 and 9. Fifteen minutes equals one-fourth of an hour. Using the same equation to find the extra location of the hand for the hour.
\[\Rightarrow 240^{\circ} + (\frac{1}{4} hr)(30)\]
\[\Rightarrow 240^{\circ} + (7.5^{\circ})\]
\[\Rightarrow 247.5^{\circ}\]
We are searching for the angle between minute hand and hour hand. The discrepancy between the two angle measurements would be equal to that.
\[\Rightarrow 247.5^{\circ} - 90^{\circ} = 157.5^{\circ}\]
3. What time would the minute-and-hour hand be at the right angle between 10 and 11?
Solution:
The time hand covered (10 * 30°) = 300° at 10 o'clock.
Two right angles will be present (clockwise and anti-clockwise)
In order to build a 90-degree angle with the hour hand, the minute hand must be at 1 or 7, provided that the hour hand is at 10.
The minute hand has to cover a proportional distance of (1 * 30) = 30° for the first right angle.
The minute hand has to cover a relative distance of (7*30) = 210° for the 2nd right angle.
We understand that the relative velocity between the two hands is 5(½)° per minute.
The time needed for the 1st right angle = \[\frac{30 \times 2}{11} = \frac{60}{11} = or 5 \frac{5}{11}\] minutes, therefore.
The time required =\[\frac{210 \times 2}{11} = \frac{420}{11} = 38 \frac{2}{11}\] minutes for the 2nd right angle.
Did You Know?
The two hands coincide once every hour. They will coincide 11 times within 12 hours. It occurs between 12 and 1'o clock because of only one such incident.
When the hands are in opposite directions, they are separated by 30 minutes of space.
By displaying a shadow's position on a flat surface, a sundial shows the time.
When the two hands are at right angles, they are separated by 15 minutes of space.
In order to measure time, an hourglass, also known as a sandglass or sand timer, uses sand trickling slowly.
If both the hour-hand and minute-hand move at their normal speeds, after \[65 \frac{5}{11}\] minutes, both hands meet.
FAQs on Clocks Angles
1. Using the Angle Between Hands of a Clock Formula Find the Angle Between Clock Hands When an Analog Clock Reads 3:15?
Ans: In the given clock angle problem, we are supposed to find the angle between the hands of a clock, let us understand some basic concepts before directly jumping to the answer. A clock face divides a circle into 12 sets of 30 degrees each, with 12 numbers. Around 3 and 4 o'clock, 15 minutes past 3:00 is 1/4 of the hour. Accordingly, the hand for the hour should be 1/4 of the gap between 3 and 4. 1/4 of the room at 30 degrees is 7.5 degrees. Since the hand of the hour travels 360 degrees in 12 hours, that is 30 degrees in one hour or 7.5 degrees in 15 minutes. The minute hand points to the 3 (for 15 minutes) and in 60 minutes it travels 360°. In other words, 90° in 15 minutes. The hand of the minute points to 90° and the hand of the hour points to 97.5° (measured clockwise from 12°). On the 3, the minute hand is straight and the hour hand is 7.5 degrees past it. Therefore, according to the clock angle formula, the answer is 97.5 degrees.
2. How Many Times a Day do the Minute and Hour Hands of a Clock Align with Each Other (the Angle Between them is Zero)?
Ans: A day begins at midnight, so at midnight, the first coincidence of hands ever takes place. The next coincidence will occur at around 1 'o'clock and 5 minutes now, watching the clock. Thus, one can infer that one coincidence of hands occurs every hour. The response should therefore be 24 times over 24 hours. It's not, however, the right answer and the right rationale. In 12 hours, we know that 11 coincidences will occur. One collision, therefore, will take place at:
One-collision frequency = 12 hours/111 hours
One-collision frequency = 12 x 60/111 mins
One-collision frequency = 720 mins/111 mins
One-collision frequency = 65 (5/11)
The value 65 (5/11) means that after every 65 minutes, (5/11) of a minute corresponds with the hands of a clock.
3. How Many Times a Day do the Clock's Minute and Hour Hands Build a Straight Line of 180° in a Day?
Ans: Every hour, except between 5'o clock and 6'o clock, one 180° straight line is rendered by the hands of the clock. The precise observation and examination of the watch give the impression that the hands make a straight line 180° exactly at 6 o'clock between 5 and 6 and may therefore not be the one that occurs between 5 and 6 o'clock. Instead, it is considered between 6 and 7 o'clock to be a straight line. Therefore, 180° straight lines are made by the hands of a clock 11 times in 12 hours and therefore generalized for 24 hours, the hands make 22 consecutive 180° lines in 24 hours.
