Clock Aptitude Reasoning Tricks & Problems - Finding Angle Between The Hands of a Clock Given Time

Summary

This video script offers a detailed explanation on calculating the angle between the hour and minute hands of an analog clock at different times. It covers various examples, such as 12:30, 1:20, 11:15, and 10:25, illustrating the step-by-step process to determine the angles. The script also explains how to find the shortest angle and the longer angle by subtracting from 360 degrees, providing a comprehensive guide for understanding clock angles.

Takeaways

• 🕒 The angle between the hour and minute hands of a clock can be calculated using the positions of the hands relative to the clock's numbers.
• 🔢 Each hour on the clock represents an angle of 30 degrees, as the full circle of 360 degrees is divided by the 12 hours.
• 📏 At 12:30, the minute hand is at the 6, and the hour hand is halfway between 12 and 1, creating an angle of 15 degrees from the 12 o'clock position.
• 📉 To find the angle at 12:30, add the angle from the 12 to the 6 (150 degrees) and the angle from the 12 to the halfway point between 12 and 1 (15 degrees), totaling 165 degrees.
• 📍 For the time 1:20, the hour hand is between 1 and 2, and the minute hand is at the 4, creating an angle that can be calculated by fractions of the hour.
• 📈 The position of the hour hand relative to the hour marks is found by dividing the minute value by 60, which gives the fraction of the hour passed.
• 🔄 At 1:20, the hour hand is two-thirds of the way between 1 and 2, and one-third of the way from 1 to 2, which translates to angles of 20 degrees and 40 degrees respectively.
• 📐 The total angle at 1:20 is found by adding the angles between the hour marks and the fractions of those angles, resulting in 80 degrees.
• 🕘 For 11:15, the minute hand is at the 3, and the hour hand is one-fourth of the way from 11 to 12, leading to an angle calculation involving fractions of 30 degrees.
• 🔢 Calculating the angle at 11:15 involves determining the fraction of the hour passed (one-fourth) and using it to find a partial angle from the 11 o'clock position (22.5 degrees).
• 📉 The total angle at 11:15 is the sum of the angle from the 12 to the 11 (90 degrees) and the partial angle (22.5 degrees), equaling 112.5 degrees.
• 🕒 For 10:25, the shortest angle between the hour and minute hands is found by considering the positions and calculating the smaller of the two possible angles, resulting in 162.5 degrees.

Q & A

What is the angle between the minute hand and the hour hand at 12:30?

At 12:30, the minute hand is at the 6, and the hour hand is halfway between 12 and 1. Since each hour represents 30 degrees, the angle between 12 and 1 is 30 degrees. Being halfway, the angle is 15 degrees. Adding the 150 degrees from 12 to 6 gives a total angle of 165 degrees.

How do you calculate the angle for the time 1:20 on an analog clock?

At 1:20, the minute hand is at the 4, and the hour hand is between 1 and 2. The hour hand is one-third of the way from 1 to 2. Each hour represents 30 degrees, so the angle between 1 and 2 is 30 degrees. Two-thirds of 30 degrees is 20 degrees. Adding the 30 degrees between 1 and 3 gives a total angle of 80 degrees.

What is the method to find the angle between the hour hand and the minute hand at 11:15?

At 11:15, the minute hand is at the 3, and the hour hand is between 11 and 12. The hour hand is one-fourth of the way from 11 to 12. Each hour represents 30 degrees, so the angle between 11 and 12 is 30 degrees. Three-fourths of 30 degrees is 22.5 degrees. Adding the 90 degrees between 12 and 3 gives a total angle of 112.5 degrees.

How do you determine the shortest angle between the hour hand and the minute hand at 10:25?

At 10:25, the minute hand is at the 5, and the hour hand is between 10 and 11. The hour hand is five-twelfths of the way from 10 to 11. Each hour represents 30 degrees, so the angle between 10 and 11 is 30 degrees. Five-twelfths of 30 degrees is 12.5 degrees. The angle between the hour hand and the minute hand is less than 180 degrees, which is 162.5 degrees.

What is the longest angle between the hour hand and the minute hand at 10:25?

The longest angle at 10:25 is the supplementary angle to the shortest angle of 162.5 degrees, which can be found by subtracting the shortest angle from 360 degrees. So, the longest angle is 360 - 162.5 = 197.5 degrees.

How can you find the angle between the hour hand and the minute hand if the clock says 2:40?

At 2:40, the minute hand is at the 8, and the hour hand is between 2 and 3. The hour hand is two-thirds of the way from 2 to 3. Each hour represents 30 degrees, so the angle between 2 and 3 is 30 degrees. Two-thirds of 30 degrees is approximately 20 degrees. Adding the 60 degrees between 12 and 2 gives a total angle of 80 degrees.

What is the significance of dividing the minutes by 60 when calculating the angle between the hour and minute hands?

Dividing the minutes by 60 gives you the fraction of the hour that has passed. This fraction is used to determine how far the hour hand has moved from the last hour mark towards the next one, which is essential for calculating the angle between the hour and minute hands.

Why is it important to consider the shortest angle between the hour and minute hands?

Considering the shortest angle is important because it represents the actual visual angle between the two hands on the clock face. It is often the most relevant measurement for practical purposes, such as determining the time until the next hour.

How does the position of the hour hand change as time passes?

The hour hand moves continuously as time passes, covering 30 degrees for each hour. It moves at a slower pace than the minute hand, and its position relative to the hour marks changes as the minutes increase.

Can you provide a formula to calculate the angle between the hour and minute hands at any given time?

Yes, the formula to calculate the angle between the hour and minute hands is: (hour * 30) + (minute / 2) for the hour hand, and (minute * 6) for the minute hand. The absolute difference between these two values gives the angle in degrees.

Outlines

00:00

🕒 Understanding Clock Angles at 12:30

The paragraph explains how to calculate the angle between the hour and minute hands of an analog clock at 12:30. It starts by noting that each hour represents 30 degrees (360 degrees / 12 hours). At 12:30, the minute hand is at the 6, while the hour hand is halfway between 12 and 1. The calculation involves determining the angle from the 12 to the 1 (30 degrees) and then finding half of that (15 degrees) because the hour hand is at the midpoint. Adding these gives a total angle of 165 degrees between the two hands.

05:00

🕗 Calculating Angles for 1:20 Using Fractions

This section teaches how to find the angle between the clock hands at 1:20. It emphasizes using fractions to determine the hour hand's position relative to the hour marks. The minute hand at the 4 indicates 20 minutes past the hour. The calculation involves finding two-thirds of an hour's 30-degree angle (20 degrees) and adding it to the angle between the 1 and 4 (60 degrees), resulting in an 80-degree angle between the hands.

10:03

🕘 Finding Shortest Angle at 10:25

The final paragraph focuses on calculating the shortest angle between the clock hands at 10:25. It explains that the minute hand at the 5 corresponds to 25 minutes, and the hour hand is between 10 and 11, slightly closer to 10. The calculation involves determining the fraction of the hour passed (5/12) and using it to find the angle between the hour hand and both the 10th and 11th hours. The shortest angle, which is less than 180 degrees, is found by subtracting the smaller angle (5/12 of 30 degrees) from 150 degrees (the angle between the 10th and 5th hours), resulting in a 162.5-degree angle.

Keywords

💡Analog Clock

An analog clock is a type of clock that displays the time using hands that rotate around a dial. In the video, the analog clock is the primary subject, as the script discusses how to calculate the angles between the clock's hands at different times. The video uses the example of a clock reading 12:30 to illustrate how the hour and minute hands position themselves, with the minute hand at the 6 and the hour hand between 12 and 1.

💡Angle

The angle in the context of the video refers to the degree of separation between the hour and minute hands of an analog clock. The video's main theme revolves around calculating these angles. For instance, at 12:30, the angle between the hands is calculated to be 165 degrees, which is less than 180 degrees and represents the shorter, or acute, angle between the hands.

💡Hour Hand

The hour hand on an analog clock is the shorter, thicker hand that indicates the hour. The video explains that at 12:30, the hour hand is not exactly at 12 but has moved halfway towards 1, which is why the angle between the hour and minute hands is calculated by considering the hour hand's position between the 12 and the 1.

💡Minute Hand

The minute hand is the longer, thinner hand on an analog clock that indicates the minutes. In the video, the minute hand's position at 12:30 is at the 6, directly opposite the 12, which is used to calculate the angle between it and the hour hand.

💡Degrees

Degrees are units used to measure angles. The video uses degrees to express the angles between the clock hands. For example, it states that one hour on the clock represents 30 degrees, as the entire clock face is 360 degrees and there are 12 hours.

💡Complete Revolution

A complete revolution refers to a full cycle or rotation, which is 360 degrees in the context of a clock. The video mentions that the hour hand takes 12 hours to make a complete revolution, correlating to 360 degrees.

💡Fraction

A fraction is a numerical value that represents a part of a whole. In the video, fractions are used to determine how far the hour hand has moved past a certain hour. For example, at 2:40, the hour hand is two-thirds of the way between 2 and 3, which is calculated by taking the minute value (40), dividing it by 60, and then finding the fraction of the hour hand's position.

💡Shortest Angle

The shortest angle in the context of the video is the smaller of the two possible angles between the hour and minute hands. The video demonstrates how to find the shorter angle at a given time, such as at 10:25, where the angle is calculated to be 162.5 degrees, which is less than 180 degrees.

💡Clock Face

The clock face is the circular surface of an analog clock where the hour and minute hands move. The video uses the clock face to explain the positions of the hands at different times, such as at 12:30 where the minute hand is at the 6 and the hour hand is between 12 and 1.

💡Time Calculation

Time calculation in the video refers to the process of determining the angle between the clock hands based on the current time. The video provides methods for calculating these angles, such as at 11:15, where the calculation involves determining the fraction of the hour hand's position and then finding the corresponding angle on the clock face.

Highlights

At 12:30, the minute hand is at the 6, and the hour hand is halfway between 12 and 1.

Each hour on a clock represents an angle of 30 degrees, as 360 degrees is divided by 12 hours.

The angle between the hour hand and the minute hand at 12:30 is calculated by adding 150 degrees and 15 degrees, resulting in 165 degrees.

At 1:20, the hour hand is between 1 and 2, and the minute hand is at the 4.

The hour hand's position at 1:20 is one-third of the way from 1 o'clock towards 2 o'clock.

The angle between the hour hand and the minute hand at 1:20 is found by adding 20 degrees, 30 degrees, and 30 degrees, totaling 80 degrees.

At 11:15, the minute hand is at the 3, and the hour hand is between 11 and 12.

The hour hand at 11:15 is one-fourth of the way from 11 o'clock towards 12 o'clock.

The angle between the hour hand and the minute hand at 11:15 is calculated by adding 90 degrees and 22.5 degrees, resulting in 112.5 degrees.

At 10:25, the minute hand is at the 5, and the hour hand is between 10 and 11.

The hour hand at 10:25 is five-twelfths of the way from 10 o'clock towards 11 o'clock.

The shortest angle between the hour hand and the minute hand at 10:25 is found by adding 12.5 degrees and 150 degrees, totaling 162.5 degrees.

The longer angle at 10:25 can be found by subtracting the shorter angle from 360 degrees, resulting in 197.5 degrees.

The sum of all three angles in a clock face must equal 360 degrees.

The method for calculating the angle between clock hands involves understanding the position of each hand relative to the hour markers.

The angle calculation changes depending on whether the minute hand is ahead or behind the hour hand.

For times when the minute hand is ahead of the hour hand, the shorter angle is calculated by adding the angles between the hour markers.

For times when the minute hand is behind the hour hand, the shorter angle is calculated by subtracting the smaller angle from 360 degrees.

Transcripts

the time on an analog clock

reads 12 30.

what is the angle between the minute

hand and the hour hand of the clock

so on a clock this is going to be 12

we have 3 on the right

six below

and nine

so at 12 30

the minute hand that's the long hand is

going to be at the six

and the short hand the hour hand

is not exactly at 12

because if it was exactly at 12 it would

be 12 o'clock

if it wasn't one o'clock the hour hand

would be at one

but if it's 12 30

the hour hand

which is the shorthand has to be between

12 and 1.

exactly right in the middle i'm gonna

represent the hour hand in red

so that's twelve thirty our goal is to

find the angle

between

those two hands

so how can we do so

now first it's helpful to know

the measure of one hour

so let's focus on the three

what's the angle between the two and the

three

notice that it takes 12 hours for the

hour hand to make a complete revolution

so 12 hours

correlates to 360 degrees because that's

the entire circle

so one hour is going to be 360 divided

by 12 so every hour represents an angle

of 30.

so therefore

from one o'clock

to six o'clock

that's a time period of five hours

so if one hour represents 30 degrees

then 5 hours is 5 times 30 which is 150

degrees

so therefore

from 1

all the way to six

that angle is 150 degrees

so now we need to find it from

this point

to this point this angle here

now we know from 12 to 1 it represents

30 degrees

so the red line is right between 12 and

1 because we're at 12 30 we're halfway

between 12 and 1.

so if we're halfway

we need to multiply 30 by half

so therefore the angle on the inside

is 15 degrees

so we got to add

150 and 15.

therefore the angle between the hour

hand and the minute hand is 165

which is less than 180

because if the hour hand was at 12 and

the minute hand was at 6 which is

impossible

that would be a straight line that would

be 180 but it has to be less than 180

and this answer is reasonable

here's another example

convert the time 120

to degrees

that is

find the angle between the hour hand and

the minute hand if the clock says 120

so first let's draw a picture

here's 12 this is going to be 3

6

and 9

and then here's one

two

four

and five

so if it's 120 that means the hour hand

is between

one and two

the minute hand has to be at four so i'm

going to use blue to represent the

minute hand which is the long one and

red to represent the hour hand

now if the out hand is one is between

one and two

where exactly is it

so focus on the minute hand which is at

20.

20 divided by 60

is a third

so what this means is that

the hour hand is one third away from

the one o'clock hour or the value of one

and it's two thirds away

from the second hour or from two

so that's why you want to take whatever

your minute value is and divided by 60

so you can get the fraction of how far

it is from one of the hours

and to find the distance from the other

one it's going to be one minus the

original fraction so one minus one third

will give you two thirds

so now with this information we could

find everything we need

so keep in mind

the angle between one hour is always 30

degrees

it's 360 divided by 12. so between 3 and

4 is 30 degrees and between 2 and 3 is

30.

now here is where it gets interesting

so this is when you want to use this

fraction

this angle

is two-thirds of an hour

that's the missing 40 minutes

so the angle for one hour is always

going to be 30 that's not going to

change we got to find two-thirds of 30.

30 divided by 3 is 10 times 2 is 20. so

therefore this angle is 20. now all you

need to do

is add up these three angles 20 plus 30

plus 30 is equal to 80.

so that is the angle between the hour

hand and the minute hand and that's how

you do it

so now it's your turn

try this example

let's say the time on an analog clock is

11 15.

go ahead and find the angle between the

hour hand and the minute hand

so feel free to pause the video and work

out this problem

now let's always begin with a picture so

this is 12 3

6

and 9.

the minute ham

is at 3

because it represents 15. 3 is for 15 6

is 30

9 is 45 12 is well

zero

so every increment of five

is for each number

so for example

let's put down the other numbers

so let's say if it was 1105

the minute hand will be at five

eleven ten

it will be at two

fifteen is for three twenty is four

twenty five is for five

thirty is for six 35 is for seven and so

forth 55 is for 11. this is 50

45 40 and then back to zero

so just in case you're wondering those

are the numbers that

you need to know

so the 15 points to three

now where is the hour hand

we know that the hour hand is between

eleven and twelve

but it's closer to eleven

what we need is the fraction

so take fifteen

the value of the minutes and divided by

sixty

fifteen over 60 reduces to one-fourth

so therefore

the hour hand is one-fourth its way from

the 11th hour which means that if we

subtract one by a fourth

that's four over four minus one over

four four minus one is three so

three-fourths

is between the hour hand and twelfth so

it's three-fourths it's one-fourth of

the weight going from eleven to twelve

it has three fourths left to get to

eleven and twelve

so basically

it's twenty five percent of the way

between eleven and twelfth

now that we have that let's go ahead and

calculate the angle

so every hour represents 30 degrees

now the angle between the hour hand and

12 is going to be 3 4 of 30.

so what is three-fourths of 30

30 divided by 4 is 7.5 7.5 times 3 is

22.5

so that's the missing angle

so if we add 30 three times that's 90.

so we got to add 90

plus

22.5

and so that's going to be

112.5

so that is the angle between the hour

hand and the minute hand

it's 112.5

so here's the last example

find the shortest angle

for

the time of 10 25 so find the shortest

angle between the hour hand and the

minute hand when the clock says 10 25

this is going to be 12 3

six and nine

so 25 corresponds to five

that's where the minute hand is going to

be located

now the hour hand

is between 10

and 11. so it's very close to the middle

but it's slightly closer to 10.

so there's the hour hand

now let's find the fraction of 25

divided by 60.

so twenty five over sixty

twenty five is five times five sixty is

five times twelve

so it's five over twelve

therefore between the hour hand and 10

is 5 12 of 30 degrees

and between

the hour hand and 11

that's going to be

1 minus 5 over twelve which is twelve

over twelve minus five over twelve

so that's seven over twelve

so seven twelfths of thirty degrees is

the angle between the hour hand and

the eleventh hour

so now that we have that we can find the

angle so we're looking for the angle of

the shortest

we're looking for the shortest angle

between the hour hand and the minute

hand

this side is clearly the longer angle

that's going to be more than 180

so we need to find the angle

of that side between the hour hand and

the minute hand

that's less than 180.

now we know that one hour

represents 30 degrees

so if we find the angle from the fifth

hour

to the tenth hour

that's five hours ten minus five is five

and five hours represent an angle

of 150 degrees

so now we just gotta find this angle

here

which is five twelfths of thirty

so what's five twelfths of thirty

so thirty times five is one fifty and

150 divided by 12

is 12.5 degrees

so we need to add

12.5 and 150 which will give us a final

answer of

162.5 degrees

so that is the angle well that is the

shorter angle

between

the the hour hand and the minute hand

when the clock says 10 25 by the way if

you want to find the longer angle

that is this angle

it's simply

360 minus this answer so 360 minus 162.5

that will give you the other angle of

197.5

and keep in mind this is 12.5 on the

inside

so all three of these angles have to add

up to 360.