3 Systems of Measuring Angles

Lines and angles are one of the basic concepts learned by students in geometry. You might be familiar with the term ‘degree’ associated with angles. 

Did you know that the commonly used unit degree to measure angles falls under a system called Sexagesimal System? Apart from this system, there are two more systems to measure angles.

Let’s understand the three systems of measuring angles – the sexagesimal system, the centesimal system, and the circular system.

3 Systems of Measuring Angles

If a straight line stands on another line and if the two adjacent angles thus formed are equal to one another then by geometry, each of these angles is called a right angle. This right angle forms the basis for defining the different systems for the measurement of angles.

The following three different systems of units are used in the measurement of angles :

  • Sexagesimal System ( or English System)
  • Centesimal System ( or French System)
  • Circular System

1. Sexagesimal System

In this system, a right angle is divided into $90$ equal parts and each such part is called a degree $\left( 1^{\circ} \right)$.  A degree is divided into $60$ equal parts and each such part is called a Sexagesimal Minute $ \left( 1^{’} \right)$. And a minute is further subdivided into $60$ equal parts, each of which is called a Sexagesimal Second $ \left( 1^{’’} \right)$. 

In short, 

  • $1$ right angle = $90$ degrees (or $90^{\circ}$)
  • $1$ degree (or $1^{\circ}$) = $60$ minutes ( or $60^{’}$)
  • $1$ minute ( or $1^{’}$ ) = $60$ seconds ( or $60^{’’}$)

A complete rotation describes $360^{\circ}$, which forms a full circle.

2. Centesimal System

In the Centesimal System, an angle is measured in grades, minutes and seconds. In this system, a right angle is divided into $100$ equal parts and each such part is called a Grade ($1g$).  Again, a grade is divided into $100$ equal parts and each such part is called a Centesimal Minute ($1^{‵}$). And a minute is further subdivided into $100$ equal parts, each of which is called a Centesimal Second ($1^{‶}$). 

In short,

  • $1$ right angle = $100$ grades (or, $100g$)
  • $1$ grade ( or $1g$) = $100$ minutes (or, $100^{‵}$)
  • $1$ minute (or $1^{‵}$) = $100$ seconds ( or, $100^{‶}$)

3. Circular System

In this System, an angle is measured in radians. In higher mathematics angles are usually measured in a circular system. In this system a radian is considered as the unit for the measurement of angles. A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius. 

In any circle, the angle subtended at its centre by an arc of the circle whose length is equal to the radius of the circle is called a radian.

sexagesimal system
1 Radian

Conversions of the Sexagesimal System and Centesimal System

You can convert the measure of angle from one system to another using their respective conversion formula.

1. Conversion – Sexagesimal System to Centesimal System

In sexagesimal system, $1$ right angle = $90^{\circ}$ and in centesimal system, $1$ right angle = $100g$

$=>90^{\circ} = 100g => 1^{\circ} = \frac{100}{90}g = \frac{10}{9}g$

For example, $54^{\circ} = 54 \times \frac {10}{9}g = \frac {540}{9}g = 60g$

2. Conversion – Centesimal System to Sexagesimal System

In centesimal system, $1$ right angle = $100g$ and in sexagesimal system, $1$ right angle = $90^{\circ}$

$=>100g =  90^{\circ} => 1g = \frac {90}{100}^{\circ} = \frac {9}{10}^{\circ} $

For example, $45g =  45 \times \frac {9}{10}^{\circ} = 40.5^{\circ}$

3. Conversion – Sexagesimal System to Circular System

In sexagesimal system, $1$ right angle = $90^{\circ}$ and in circular system, $1$ right angle = $\frac{\pi}{2}$ radian

$=>90^{\circ} = \frac {\pi}{2} radian => 1^{\circ} = \frac {\pi}{180} radian$ 

For example $30^{\circ} = 30 \times \frac {\pi}{180} = \frac {\pi}{6} radian$

4. Conversion – Circular System to Sexagesimal System

In circular system, $1$ right angle = $\frac {\pi}{2}$ and in sexagesimal system, $1$ right angle = $90^{\circ}$ 

$=>\frac {\pi}{2} radian = 90^{\circ} => 1 radian = \frac {180}{\pi}^{\circ}

For example, $ \frac {\pi}{4} radian = \left( \frac {\pi}{4} \right) \times \left( \frac {180}{\pi} \right) = 45°$

5. Conversion – Centesimal System to Circular System

In centesimal system, $1$ right angle = $100g$ and in circular system, $1$ right angle = $ \frac {\pi}{2}$ radian

$=>100g = \frac {\pi}{2} radian => 1g = \frac {\pi}{200} radian$

For example, $60g = 60 \times \left( \frac {\pi}{200} \right) = \left( \frac {3}{10} \right) \pi$

6. Conversion – Circular System to Centesimal System

In circular system $1$ right angle = $ \frac {\pi}{2}$ radian and in centesimal system, $1$ right angle = $100g$

$=> \frac {\pi}{2} radian = 100g => 1 radian = \frac {200}{\pi} g $

For example, $ \frac {\pi}{6} radian = \left( \frac {\pi}{6} \right) \times \left( \frac {200}{\pi} \right) = 33.33 g$

FAQs

What is meant by a sexagesimal system?

The sexagesimal system was an ancient system of counting, calculation, and numerical notation that used powers of 60 much as the decimal system uses powers of 10. Rudiments of the ancient system survive in a vestigial form in our division of the hour into 60 minutes and the minute into 60 seconds.

Where do we use the sexagesimal system?

We use the sexagesimal system in measuring angles, geographic coordinates, electronic navigation, and time. One hour of time is divided into 60 minutes, and one minute is divided into 60 seconds.

What is the centesimal system?

In the Centesimal system of angle measurement, the right angle is split into 100 equally divided parts known as Grades. Each grade is split into 100 minutes and every single minute is 100 seconds. 1 right angle = 100g (100 grades) 1 grade = 100” (100 minutes) 1 minute = 100′ (100 seconds)

Where do we use the centesimal system?

The Centesimal System is another measurement system to measure angles in trigonometry. In this system, a Right Angle is divided into 100 equal parts known as Grades. Also in this system, each grade is subdivided into 100 equal parts called Minutes, and each minute is subdivided into 100 equal parts known as Seconds

Conclusion

The three systems of measurement of angles are suitable for specific applications and fields. The Sexagesimal system is the most commonly used system in lower fields of mathematics. The centesimal system of angle measurement did not receive much recognition because of its complexity. The circular system is used in higher fields of study and trigonometry.

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