# Problems Based on Systems of Measuring Angles

Problems based on systems of measuring angles will help us to learn converting one measuring systems to other measuring systems. We know, the three different systems are Sexagesimal System, Centesimal System and Circular System. The examples will help us to solve various types of problems involving the three different systems of measuring angles.

Worked-out problems based on systems of measuring angles:

1. 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.

2. Two regular polygons have sides m and n respectively. If the number of degrees in an angle of the first is equal to the number of grades in an angle of the second, show that,

20/n - 18/m = 1.

Solution:

Sum of the internal angles of a regular polygon of m sides = (2m - 4) rt. angles.

Therefore, one angle of a regular polygon of m sides measures (2m - 4)/m rt. angles.

Similarly, one angle of a regular polygon of n sides measures (2n - 4)/n rt. angles.

By question, [(2m - 4)/m]  × 90 = [(2n - 4)/n] × 100

[Since, 1 rt. angle = 90° = 100g]

or, (1 - 2/m)  × 180 = (1 - 2/n) × 200

or, 9 - 18/m = 10 - 20/n

or, 20/n - 18/m = 1.  Proved

Measurement of Angles

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