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Clocks and Angles

A clock is included among the most oldest human inventions. It is an important part of our lives. The clock is said to be a machine that keeps us indicated about the time. The term "clock" was assumed to be taken from Celtic words "clocca" or "clagan" which mean "bell". A more silent clock (very low tick-tock sound) is usually known as the "timepiece". This article will throw light on the mathematics behind the concept of a clock. We will discuss the angles formed by the minute and hour hands of clocks in detail.

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Generally, a clock is defined as an instrument that measures and displays time. While wristwatches and timepieces are different from a clock. Generally, we use two types of clocks - digital and analog. Digital clock is the one which displays time using digits while an analog clock has hour and minute hands. Since we are focusing on angles of the clock, hence by the term clock, here we will refer to analog clock only.

A clock is divided into 12 parts. Any two parts are inclined at an angle measuring $30^{\circ}$. It is shown in the following diagram :

Clocks and Angles

Facts About Analog Clock

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The analog clocks use angles to indicate the time. Analog clocks have moving hands and numbered dial. Ideally, it has a 12-hours circular scale. This scale has a 60-minute scale and an even 60-second scale too, in the case of the clock having a second hand. However, in several different designs and styles of analog clocks, there may be different types of dials or scales. But the most commonly used dial is the 12-hour dial. The general facts about analog clocks are :

(i) The most basic analog clock's face is a circular dial which has numbers from 1 to 12.

(ii) Each number corresponds to an hour.

 The whole dial is sectioned in 60 small marks, known as the minute marks. The hour marks are included in them.

There is a 60$^{\circ}$ angle between every two adjacent numbers or marks on an analog clock dial.

Angle Between Minute and Hour Hand

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We know that a clock is divided into 12 hours using 12 marks. An hour is equal to 60 minutes. The minute hand completes its full circle in 60 minutes completing 360$^{\circ}$ at a constant speed. Complete turn makes 360 angle, so $\frac{360^{\circ}}{12}$ = $30^{\circ}$ is the angle between hands of a clock. Also, there are 5 small marks between any two adjacent hour marks. The angle between two small marks is 6$^{\circ}$

Let's take an example of finding the angle between minute and hour hand of a clock displaying 2:20. 

Angle Between Minute and Hour Hand

The angle of hour hand is given as 70$^{\circ}$. The angle made by minute hand is 30 $\times$ 4 = 120$^{\circ}$. So, the angle between minute and hour hand is 120$^{\circ}$ - 70$^{\circ}$ = 50$^{\circ}$. Also, the reflex angle is equal to 360$^{\circ}$ - 50$^{\circ}$ or 310$^{\circ}$.

Clock Angle Formula

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Hour Hand Angle
The formula for angle made by hour hand is given below.
$\alpha = 0.5^{\circ} \times (60 \times H + M)$

Where $\alpha$ = Angle made by hour hand clockwise from 12 (degrees)
H = Number of hours
M = Number of minutes past given hour.

Minute Hand Angle
The formula for angle made by minute hand is illustrated below.
$\beta = 6^{\circ} \times M$
where, $\beta$ = Angle made by minute hand clockwise from 12 (degrees)
M = Number of minutes past hour.

Angle Between Hands
The formula for angle between hands is :
$\theta = |\alpha - \beta|$


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Have a look at following examples.

Example 1 : Find the hour hand and minute hand angle when the clock shows a time of 3:35.

Solution :
 Here. H = 3 and M = 35

Hour hand angle

$\alpha = 0.5^{\circ} \times (60 \times H + M)$

$\alpha = 0.5^{\circ} \times (60 \times 3 + 35)$

= $0.5^{\circ} \times 215$

= 107.5$^{\circ}$

Minute hand angle

$\beta = 6^{\circ} \times M$

= $6^{\circ} \times 35$

= 210$^{\circ}$
Example 2 : Calculate the angle between hour hand and minute hand at quarter to 9.

Solution :
 Quarter to 9 is 8:45.

Here. H = 8 and M = 45

$\alpha$ and $\beta$ are angles of the hour hand and minute hand respectively.

Hour hand angle

$\alpha = 0.5^{\circ} \times (60 \times H + M)$

$\alpha = 0.5^{\circ} \times (60 \times 8 + 45)$

= $0.5^{\circ} \times 525$

= 262.5$^{\circ}$

Minute hand angle

$\beta = 6^{\circ} \times M$

= $6^{\circ} \times 45$

= 270$^{\circ}$

Angle between minute and hour hand

Let $\theta$ be the angle between hour and minute hand,

$\theta = |\alpha - \beta|$

= $|262.5 - 270|$

= 7.5$^{\circ}$
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