What’s the angle between the clock hands when the time is 3:15?

The clock face is a circle and a circle has a total of 360 degrees all the way around its centre.

Each number on the clock face represents an hour and is evenly spaced out by five minute intervals. This gives us a total of 12 equal parts.

To get the angle between the clock hands when they are exactly at five minute intervals, we divide 360 by 12.

360 ÷ 12 = 30°

Therefore when the time is 1:00 the angle will be 30°.

To find the angle between the clock hands when the time is 3:00, we divide 360 by 12 and then multiply the answer by 3.

(360 ÷ 12) × 3 = 30 × 3 = 90°

Similarly at 6:00 the angle will be:

(360 ÷ 12) × 6 = 30 × 6 = 180°

When the time is 3:15 both the hands do not rest on number 3. Though the minute hand will be on the number 3 but the hour hand would have moved a quarter of distance between numbers 3 and 4 because 15 minutes is also known as a quarter or one-fourth (1/4) of an hour.

Therefore to get the angle between the clock hands when the time is 3:15, we further divide the five minute angle by 4.

(360 ÷ 12) ÷ 4 = 30 ÷ 4 = 7.5°

What will be the angle if the clock hands are at 8:00?

Let’s calculate.

(360 ÷ 12) × 8 = 30 × 8 = 240°

To get the angle between the clock hands, subtract it from 360

360 - 240 = 120°

Why did we do this?

Any angle greater than 180° but less than 360°, known as a reflex angle, has to be subtracted from 360 to get the value of the smaller angle.

At 12:00 both hands are on the number 12. Hence the angle will be

(360 ÷ 12) × 12 = 30 × 12 = 360°

But this is greater than 180°. Therefore taking away 360 from it, we get:

360 - 360 = 0 degree

But what if we have to find the angle at 4:12 or 7:56?

To make such calculations easier we will have to come up with a general purpose solution.

To proceed we have to calculate the angles of the hour's hand and minute's hand separately with respect to 12 o’clock and then subtract them to find the required angle.

The angle between the two hands = Angle of hour's hand - Angle of minute's hand

To find the angle of the minute's hand we first divide 360 into 60 equal parts because we know that there are 60 minutes in an hour and an hour is one whole circle on the clock face.

Angle of minute's hand = (360 ÷ 60) × position of minute's hand

To get the angle of hour's hand we first divide 360 into 12 equal parts because we know that the clock face is divided into 12 hours.

(360 ÷ 12) × position of hour's hand

But we know that the hour's hand can be between two numbers. To fine tune our calculation we have to find this additional angle in terms of hours.

((360 ÷ 60) × position of minute's hand) ÷ 12

Which is nothing but —

(Angle of minute's hand) ÷ 12

And add these to get the angle of hour's hand.

∴ Angle of hour's hand = ((360 ÷ 12) × position of hour's hand) + (Angle of minute's hand ÷ 12)

And if the angle of minute's hand is greater than the angle of hour's hand, we have to drop the minus sign, i.e. take the absolute value of their difference.

Additionally, after subtracting, if we get an angle greater than 180°, we know that it’s a reflex angle. Therefore we will again subtract it from 360 to get the measure of the smaller angle between the two hands.

Let’s try it.

For the time 3:15

Angle of minute's hand = (360 ÷ 60) × 15 = 6 × 15 = 90°

Angle of hour's hand = ((360 ÷ 12) × 3) + (90 ÷ 12) = 90 + 7.5 = 97.5°

Angle between the hands at 3:15 = 97.5 - 90 = 7.5°

Eureka! It works.

For the time 4:12

Angle of minute's hand = (360 ÷ 60) × 12 = 6 × 12 = 72°

Angle of hour's hand = ((360 ÷ 12) × 4) + (72 ÷ 12) = 120 + 6 = 126°

Angle between the hands at 4:12 = 126 - 72 = 54°

For the time 7:56

Angle of minute's hand = (360 ÷ 60) × 56 = 336°

Angle of hour's hand = ((360 ÷ 12) × 7) + (336 ÷ 12) = 210 + 28 = 238°

Angle between the hands at 7:56 = 238 - 336 = |-98| = 98° (Taking the absolute value.)

One last example for the time 10:06

Angle of minute's hand = (360 ÷ 60) × 6 = 6 × 6 = 36°

Angle of hour's hand = ((360 ÷ 12) × 10) + (36 ÷ 12) = 300 + 3 = 303°

Angle between the hands at 10:06 = 303 - 36 = 267°

Angle is greater than 180°

Therefore smaller angle between the hands at 10:06 = 360 - 267 = 93°

Now let’s write an algorithm (a recipe) to find the angle between the two hands at any given time.

  1. Input hour
  2. Input minute
  3. AngleMinute = (360 ÷ 60) × minute
  4. AngleHour = ((360 ÷ 12) × hour) + (AngleMinute ÷ 12)
  5. Angle = | AngleHour - AngleMinute |
  6. If Angle > 180 then Angle = 360 - Angle
  7. Output Angle

Since we know that a circle is always going have 360 degrees, and there are 60 minutes in a hour and 12 is the number of hours marked on the clock face, we can replace the calculations involving these constants with their answers. Hence our algorithm becomes:

  1. Input hour
  2. Input minute
  3. AngleMinute = 6 × minute
  4. AngleHour = (30 × hour) + (AngleMinute ÷ 12)
  5. Angle = | AngleHour - AngleMinute |
  6. If Angle > 180 then Angle = 360 - Angle
  7. Output Angle

From the algorithm given above, can you figure out the one minute angle?

By Amar Preet
May 2020