Love thy Number: Angle between the Hands

Beauty of Mathematics

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry-- Bertrand Russel.

Tuesday, 9 June 2026

Angle between the Hands

Let us make a formula to know the exact angle between the minute hand and hour hand of a Clock. Let the angle between the hands be A, no. of hours be H and no. minutes be M.

So we will establish a relationship between A,H and M.

First of all let us determine angular speeds of minute hand and hour hand.

Minute hand travels 360 degrees in 60 minutes. Hence its speed is 360/60 or 6° per minute.

The hour hand being much slower, takes 12 hours or 720 minutes to cover 360 degrees.

Hence hour hand’s speed = 1/2° per minute or 30° per hour.

Imagine a clock initially showing 0° angle, which occurs exactly at 12 O’ Clock. And also imagine a vertical line between 12 O’ Clock and 6 O’ Clock.

Suppose, the hour hand travels for H complete hours and M Minutes from the initial position, and M hand travels for M minutes from the initial position.

Angular distance covered by Minute Hand = speed x time =6 M

Angular distance covered by Hour Hand = H hours + M minutes= 30H+ M/2.

Now the angle between the hour hand and minute hand will be the difference of the two angular distances

Please note that for an angle A, sometimes minutes hand will be ahead and sometimes hour hand will be ahead.

Hence the angle A will be equal to = 6M- ( 30H +M/2)= 11M/2-30H ( This will happen when Minute hand will be ahead)

A= 11M/2-30H

And when Hour hand will be ahead the angle A will be equal to 30H+ M/2 -6M= 30 H-11M/2

A= 30 H-11M/2

We can now write the result in the modulus form, combined for both the cases:

A = |11M/2-30H|

In the future you can just put the values in the above formula and find the unknown quantity.

Even if you don't understand the derivation of the formula, you can still use it smartly.

For example look at the problem given in the figure:

Here hour hand is between 9 and 10. Hence H= 9.

Angle A= 180°.

As the hour hand is ahead, we will use the second formula.

A= 30 H-11M/2

Putting the values in the above equation, we can find the value of M.

180= 30X9-11M/2

M= 180/11.

M = 16 and 4/11 minutes.

Normally such answers are conventionally given in the mixed fraction form only. However the same can be converted into seconds also.

So the final answer will be 16 minutes, 21 and 2/11 seconds.

Or the same may be written as 16 minutes and 21.8181 seconds or 16 minutes and 21.82 seconds. You can say approximately 16 minutes and 22 seconds also.