Systems of Measuring Angles

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

(a) Sexagesimal System ( or English System)

(b) Centesimal System ( or French System)

(c) Circular System

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 in defining the different systems for the measurement of angles.


Definition of systems of measuring angles:

(a) Sexagesimal System: In Sexagesimal System, an angle is measured in degrees, minutes and seconds.

A complete rotation describes 360°. In this system, a right angle is divided into 90 equal parts and each such part is called a Degree (1°); a degree is divided into 60 equal parts and each such part is called a Sexagesimal Minute (1’) and a minute is further sub-divided into 60 equal parts, each of which is called a Sexagesimal Second (1’’). In short, 

1 right angle

1 degree (or 1°)

and 1 minute ( or 1’ )

= 90 degrees (or 90°)

= 60 minutes ( or 60’)

= 60 seconds ( or 60’’).

(b) Centesimal System: In 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 sub-divided into 100 equal parts, each of which is called a Centesimal Second (1‶). In short,

1 right angle
1 grade ( or 1g)
and 1 minute (or 1‵)
= 100 grades (or, 100g)
= 100 minutes (or, 100‵)
= 100 seconds ( or, 100‶).

Note: (i) Clearly, minute and second in sexagesimal and centesimal systems are different.

For example,

1 right angle = 90 × 60 = 5400 sexagesimal minutes = (5400)’

and 1 right angle = 100 × 100 = 10000 centesimàl minutes = (10000)‶

(ii) Since, 1 right angle = 90° = 100g
Therefore, 90° = 100g
or, 1° = (10/9) g and 1g = (9/10)°

The first relation is used to reduce an angle of sexagesimal system to centesimal system and the second is used to reduce an angle of centesimal system to sexagesimal system.

(c) Circular System: In this System, an angle is measured in radians. In higher mathematics angles are usually measured in circular system. In this system a radian is considered as the unit for the measurement of angles.

Definition of Radian: A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius.


A radian defined as follows:

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. Let OX = r be the radius of a circle having center at O.
A Radian
Now, take an arc XY of the circle such that arc XY = r and join OY. By definition, ∠XOY = one radian. One radian is written as 1c, 2 radians as 2c and in general, k radians as kc.

Circular (radian) measure of an angle:

The circular measure of an angle is the number of radians it contains.

Thus the circular (radian) measure of a right angle is π/2.

If an angle is given without mentioning units, it is assumed to be in radians. The relation between degree measures and circular (radian) measures of some standard angles are given below:

Degrees

30°

45°

60°

90°

120°

135°

150°

180°

270°

360°

Radians

0

π/6

π/4

π/3

π/2

2π/3

3π/4

5π/6

π

3π/2


Measurement of Angles





11 and 12 Grade Math

From Systems of Measuring Angles to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. 4th Grade Mental Math on Roman Numerals | Roman Numerals Quiz

    Feb 23, 24 03:55 PM

    In 4th grade mental math on numbers, students can practice different questions on write the Hindu-Arabic numerals, write the Roman Numerals, comparison of roman numerals, addition of roman numerals.

    Read More

  2. 4th Grade Mental Math on Numbers | Mental Math 4th Grade with Answers

    Feb 23, 24 02:24 PM

    4th Grade Mental Math on Numbers
    In 4th grade mental math on numbers, students can practice different questions on numbers in figures, number name, place value, face value, comparison of number and formation of greatest and smallest…

    Read More

  3. Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

    Feb 23, 24 01:28 PM

    List of Roman Numerals Chart
    How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

    Read More

  4. Worksheet on Roman Numerals |Roman Numerals|Symbols for Roman Numerals

    Feb 22, 24 04:15 PM

    Roman Numbers Table
    Practice the worksheet on roman numerals or numbers. This sheet will encourage the students to practice about the symbols for roman numerals and their values. Write the number for the following: (a) V…

    Read More

  5. Roman Symbols | What are Roman Numbers? | Roman Numeration System

    Feb 22, 24 02:30 PM

    Roman Numbers
    Do we know from where Roman symbols came? In Rome, people wanted to use their own symbols to express various numbers. These symbols, used by Romans, are known as Roman symbols, Romans used only seven…

    Read More