Measurement of Angles
Angles
Angles are the shape formed by two lines or rays which are diverging from a common point or the vertex.
Here, AB is the initial line and AC is a revolving line. Here, AC starts from AB, which revolves in the anticlockwise (opposite) direction, then the angle ABC is called positive angle.
If AC starts from AB and revolve round A, it is clockwise (positive) direction, then the angle ABC is called negative angle.
Measurement of Angles
There are three systems which are commonly used for the measurement of angles and they are listed below:
 Sexagesimal system or English system (Degree system)
 Centesimal system or French system (Grade system)
 Radian system or Circular measure
(NOTE: The Right angle is taken as the standard angle in each of these systems.)
Sexagesimal system or English system
It is a system where the unit of measurement of an angle is degree. In this system, the right angle is divided into 90 equal parts and each part is called a degree. It is a base with 60 which uses the concept of degrees, minutes and seconds for measuring angles where,
1 right angle = 90° (90 degrees)
1° = 60' (60 sexagesimal minutes)
1' = 60" (60 sexagesimal seconds)
Centesimal system or French system
It is a system where the unit of measurement of an angle is grade. In this system where the unit of measurement of an angle is grade. In this system, a right angle is divided into two equal parts and each part is known as a grade.It uses the concept of grade, minute and seconds where,
1 right angle = 100^{g} (100 grades)
1 = 100' (100 centesimal minutes)
1' = 100" (100 centesimal seconds)
Radian system or Circular measure
Radian is the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. In this figure O is the centre of a circle and OA is the radius. Let's take point B on the circumference making OA = arc AB and join OB. The angle AOB hence formed and called1 radian. In this measure, 1 right angle = (\(\frac{π}{2}\) )^{c}
^{}
Relation between different systems of measurement
Relation between Sexagesimal and Centesimal system: 
As we know that,
1 right angle = 90°
1 right angle = 100^{g}
So, 90° = 100^{g}
or, 1° = (\(\frac{100}{90}\))^{g}
= (\(\frac{10}{9}\))^{g}
so, x° = (\(\frac{10}{9}\)×x)°
Again
100^{g} = 90°
or, 1^{g} = (\(\frac{90}{100}\))°
or, 1^{g}= (\(\frac{9}{10}\))°
So, x^{g} = (\(\frac{9}{10}\)×x)°
Relation between Sexagesimal and Radian system
As we know that,
180° =π^{c}
or, x° = (\(\frac{π}{180}\))
Similarly,
1^{c} = (\(\frac{180}{π}\))°
So, x^{c} = (\(\frac{180}{π}\)×x)°
Relation between Centesimal and Radian system
We have,
200^{g}= π^{c}
or, x^{g} = (\(\frac{π×x}{200}\))^{c}
Again
π^{c}= 200^{g}
or, 1^{c} = (\(\frac{200}{π}\))^{c}
So, (\(\frac{200×x}{π}\))^{c}
^{}
Polygon
Polygon is a closed plane figure having three or more than three line segments. Triangles, quadrilateral, nonagon, octagon etc. are the examples of a polygon. A polygon having all sides equal in length is called a regular polygon. A regular polygon has the same measures of interior angles.
In regular polygons of sides n, each interior angle is θ = \(\frac{(n2) × 180°}{n}\)
Similarly,
The exterior angle is a side of a regular polygon where an angle between any side of a shape and a line extended from the next side.
Exterior angle of polygon (Φ) = \(\frac{360°}{n}\)
Some types of polygon are discussed below: 
Polygons  No. of sides 
Triangle  3 
Quadrilateral  4 
Pentagon  5 
Hexagon  6 
Heptagon  7 
Octagon  8 
Nonagon  9 
Decagon  10 
 Angles are the shape formed by two lines or rays which are diverging from a common point or the vertex.
 In the Sexagesimal system, the right angle is divided into 90 equal parts and each part is called a degree.
 In this system, a right angle is divided into two equal parts and each part is known as a grade.
 Radian is the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc.
= (30 × 3600)" + (30 × 60)" + 30"
= 108000" + 1800" + 30"
= 109830"
= 18° + (\(\frac{30}{60}\))° + (\(\frac{15}{60×60}\))°
= 18°+ (\(\frac{30}{60}\))°+ (\(\frac{15}{3600}\))°
= (\(\frac{18×3600 + 60×30 + 15}{3600}\))°
= (\(\frac{66615}{3600}\))°
= 18.5042°
= (50 × 10000)" + (49 × 100)" + 38"
= 500000" + 4900" 38"
= 504938"
= 50^{g} + (\(\frac{87}{100}\))^{g} + (\(\frac{50}{100×100}\))^{g}
= 50^{g} + (\(\frac{87}{100}\))^{g} + (\(\frac{50}{10000}\))^{g}
= 50^{g} + 0.87^{g} + 0.005^{g}
= (50+0.87+0.005)^{g}
= 50.875^{g}
Here,
x = 72°
we have,
or, x° = (\(\frac{10}{9}×x\))^{g}
or, 72° = (\(\frac{10}{9}×72\))^{g}
∴ 72° = 80^{g}
Here,
or, x = 60°
we have,
or, x^{g}= (\(\frac{9}{10}×x\))°
or, 60^{g} = (\(\frac{9}{10}×60\))°
∴, 54°
Here,
or, x = 60°
we have,
or, x° = (\(\frac{π}{180}×x\))^{c}
or, 60° = (\(\frac{π}{180}×60\))^{c}
= \(\frac{π^c}{3}\)
Here,
or, x = 75^{g}
we have,
or, x° = (\(\frac{π}{200}×x\))^{c}
∴, 75^{g}=(\(\frac{π}{200}×75\))^{c}
^{}= \(\frac{3π^c}{8}\)
Here,
or, x = \(\frac{2π^c}{5}\)
we have,
or, x^{c} = (\(\frac{180}{π}×x\))°
∴ \(\frac{2π^c}{5}\) = (\(\frac{180}{π}\) × \(\frac{2π}{5}\))°
= 72°
Here,
or, x = \(\frac{4π^c}{5}\)
we have,
or, x^{c} = (\(\frac{200}{π}×x\))^{g}
∴ \(\frac{4π^c}{5}\) = (\(\frac{200}{π}\) ×\(\frac{4π}{5}\))^{g}
= 160^{g}
or, 27° 30' = 27° + (\(\frac{30}{60}\))°
= 27° + (\(\frac{1}{2}\))°
= (\(\frac{55}{2}\))°
∴ 27° 30' = (\(\frac{55}{2}\) × \(\frac{10}{9}\))^{g}
= (\(\frac{275}{9}\))^{g}
Here,
or, 42^{g}60' = 42^{g}+ (\(\frac{60}{100}\))^{g}
= 42^{g}+ (\(\frac{3}{5}\))^{g}
= (42.6)^{g}
= (\(42.6 × \frac{9}{10}\))°
= 38.34°
Here,
\(\angle\)A = 60°, \(\angle\)B = 90°
Now,
\(\angle\)A + \(\angle\)B + \(\angle\)C = 180°
or, 60° + 90° + \(\angle\)C = 180°
or, \(\angle\)C = 180° − 150°
or, \(\angle\)C = 30°
Then,
or, 1° = (\(\frac{10}{9}\))^{g}
or, \(\angle\)A = 60° = (\(60×\frac{10}{9}\))^{g} = (\(\frac{200}{3}\))^{g}
or, \(\angle\)B = 90° = (\(90×\frac{10}{9}\))^{g} = 100^{g}
or, \(\angle\)C = 30° = (\(30×\frac{10}{9}\))^{g} = (\(\frac{100}{3}\))^{g}
solution:
In 60 minutes, the minutes hand makes 360°
In 1 minute, the minute hand makes (\(\frac{360}{60}\))°In 15 minutes, the minute hand makes (\(\frac{360×15}{60}\))° = 90°
Hence, a minute hand makes 90° in 15 minutes.
Solution:
Here,
x = 100
we have,
or, x^{g} = (\(\frac{π}{200} × x \))^{c}
or, 100^{g}= (\(\frac{π}{200} × 100 \))^{c}∴ 100^{g}= (\(\frac{π}{2}\))^{c}

What is the result, when we convert 27° 30' into grade?
((frac{274}{6}))g
((frac{246}{2}))g
((frac{275}{9}))g
((frac{245}{4}))g

What is the result, when we convert 42° 60' into degree?
39.28°
35.46°
38.34°
34.57°

What is the result, when we convert 50^{g} 49' 38" into centesimal second?
;i:1;s:7:
504938
503333

What is result, when we convert 50^{g} 87' 50" into sexagesimal measure
50.875g
51.534g
54.6363g
54.638g

What is the result, when we convert 72° into garde?
80g
78g
56g
88g

What is the result, when we convert 60^{g }into degree?
35°
26°
67°
54°

What is the result, when we convert 60° into radian measure?
(frac{π^c}{9})
(frac{π^c}{7})
(frac{π^c}{5})
(frac{π^c}{3})

What is the result, when we convert 75^{g} into radian measure?
(frac{3π^c}{8})
(frac{2π^c}{4})
(frac{8π^c}{3})
(frac{7π^c}{8})

What is the, result, when we convert (frac{2π^c}{5}) into degree?
53°
43°
93°
72°

What is result, when we convert (frac{4π^c}{5}) into grade?
106g
160g
170g
150g

What is the result, when we convert 30° 30' 30" into sexagesimal second?
163736
;i:1;s:7:
109830

What is the result, when we convert 18° 30' 15" into degree?
18.5042°
81.2387°
19.2525°
32.7098°

What is the result, when we convert 100^{g} into radian measure?
((frac{π}{4}))c
((frac{π}{2}))c
((frac{π}{6}))c
((frac{π}{8}))c

What is the result, when we convert 230^{g} into degree?
207°
206°
204°
205°

What is the value of π ?
(frac{23}{6})
(frac{7}{22})
(frac{21}{7})
(frac{22}{7})

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Discussions about this note
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Prashansa JhaAll of sexagesimal 
Mar 24, 2017 
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Amit ThapaIf the unite of angular measurement is 1/9 of a right angle, what is the measure of 34° ? 
Mar 22, 2017 
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