Sexagesimal, Centesimal and Circular Systems

We know, Sexagesimal, Centesimal and Circular Systems are the three different systems of measuring angles. Sexagesimal system is also known as English system and centesimal system is known as French system.

To convert the one system to the other system its very necessary to know the relation among the Sexagesimal system, Centesimal system and Circular system.

The relation between Sexagesimal, Centesimal and Circular systems are discussed below:

Since 90° = 1 right angle, hence, 180° = 2 right angles.
Again, 100g = 1 right angle; hence, 200g = 2 right angles.
And, πc = 2 right angles.
Therefore, 180° = 200g = πc.



Let, D°, Gg and Rc be the sexagesimal, centesimal and circular measures respectively of a given angle.
Now, 90° = 1 right angle
Therefore, 1° = 1/90 right angle
Therefore, D° = D/90 right angle

Again, 100g = 1 right angle
Therefore, 1g = 1/100 right angle
Therefore, Gg = G/100 right angle.

And, 1c = 2/π right angle
Therefore, Rc = 2R/π right angle.

Therefore we have,
D/90 = G/100 = 2R/π
               or,
D/180 = G/200 = R/π


1. The circular measure of an angle is π/8; find its value in sexagesimal and centesimal systems.

Solution:

πc/8

= 180°/8, [Since, πc = 180°)

= 22°30'

Again, πc/8

= 200g/8 [Since, πc = 200g)

= 25g

Therefore, the sexagesimal and centesimal measures of the angle πc/8 are 22°30' and 25g respectively.


2. Find in sexagesimal, centesimal and circular units an internal angle of a regular Hexagon.

Solution:

We know that the sum of the internal angles of a polygon of n sides = (2n - 4) rt. angles.

Therefore, the sum of the six internal angles of a regular pentagon = (2 ×  6 - 4) = 8 rt. angles.

Hence, each internal angle of the Hexagon = 8/6 rt. angles.  =  4/3 rt. angles.

Therefore, each internal angle of the regular Hexagon in sexagesimal system measures 4/3  ×   90°,  (Since, 1 rt. angle = 90°) = 120°;

In centesimal system measures

4/3 × 100g (Since, 1 rt. angle = 100g)

= (400/3)g

= 1331/3

and in circular system measures (4/3 × π/2)c, (Since, 1 rt. angle = πc/2)

= (2π/3)c.


3. The angles of a triangle are in A. P. If the greatest and the least are in the ratio 5 : 2, find the angles of the triangle in radian.

Solution:

Let (a - d), a and (a + d) radians (which are in A. P.) be the angles of the triangle where a> 0 and d > 0.

Then, a - d + a + a + d = π, (Since, the sum of the three angles of a triangle = 180° = π radian)

or, 3a = π

or, a =  π/3.

By problem, we have,

(a + d)/(a – d) = 5/2

or, 5(a – d) = 2(a + d)

or, 5a - 5d = 2a + 2d.

or, 5a – 2a = 2d + 5d

or, 3a = 7d

or, 7d = 3a

or, d = (3/7)a

or, d = (3/7)  × ( π/3)

or, d = π/7

Therefore, the required angles of the triangle are (π/3- π/7), π/3 and (π/3 + π/7) radians

i.e.,  4π/21, π/3 and 10π/21 radians.

Measurement of Angles





11 and 12 Grade Math

From Sexagesimal Centesimal and Circular Systems 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