Angle Measurement

What is Angle?

When a ray OA starting from its initial position OA rotates about its endpoint o and takes final position OB

When angle is formed

Positive and Negative Angles

An angle formed by a rotating ray is said to be positive or negative depending on whether its moves anticlockwise or clockwise direction respectively.

Measurement of Angle

Sexagesimal System / Degree Measure

This is also called an English system.

In this system,

1st right angle = 90o

1o = 60’

1’ = 60’’

Centesimal system of Angle Measurement

This is also known as French system.

1=1001=100{1}’ = {100}’\\ {1}’ = {100}”.

Circular system of Angle Measurement

This is very popularly known as radian system.

In this system, the angle is measured in radian

  • A radian is an angle subtended at the centres of a circle by an arc, the whole length is equal to the radius of the circle.

Note:

The numbers of radians in an angle subtended by an arc of the circle at the centres is equl to arc/ radius

θ=arcradius\theta =\frac{arc}{radius}

Important Conversions:

  1. πradian=180\pi \,radian=180{}^\circ
  2. 1radian=180π1\,radian=\frac{180}{\pi }
  3. 1=(π180)1{}^\circ =\left( \frac{\pi }{180} \right) radian
  4. If D is number of degree, R is the number of radians and G is the number of grade in angle θ\theta , then D90=G100=2Rπ\frac{D}{90}=\frac{G}{100}=\frac{2R}{\pi }
  5. θ=1r\theta =\frac{1}{r} Where θ\theta= angle sustended by arc of length 1 at centre of circle .
  6. π radian = 180°
  7.  Some of the standard radian to degree conversions are given below:
  • π/6 Radian = 30°
  • π/4 Radian = 45°
  • π/3 Radian = 60°
  • π/2 Radian = 90°
  • 2π/3 Radian = 120°
  • 3π/4 Radian = 135°
  • 5π/6 Radian = 150°
  • 7π/6 Radian = 210°
  • 5π/4 Radian = 225°
  • 5π/3 Radian = 300°

 

Related Videos:

Different Types of Angles

Draw Angles

 

Practice Problems

Illustration 1:

Write (3.25) oin D-M-S

Solution: 3o– 0.25 X 60’ = 3o15’

Illustration 2:

Write in (12.3456)g G- M- S

Solution: 12g3456{{12}^{g}}-34′-56”

Illustration 3:

Correct in to radian 30o

Solution: 30Xπ180=π630{}^\circ X\,\frac{\pi }{180}=\frac{\pi }{6}

Remember

πc=180{{\pi }^{c}}=180 1c=180π=18022X7=571622{{1}^{c\,}}=\frac{180}{\pi }=\frac{180}{22}X7\,=\,57{}^\circ 16’22” 1c57{{1}^{c\,}}\simeq 57{}^\circ

Illustration 4:

If the angles of a triangle are in the ratio 1 : 2 : 3, then find the angles in degrees.

Solution:

Let the angles be x, 2x and 3x.
Then, x + 2x + 3x = 180°    ….[∵ sum of the angles of a triangle = 180°] ∴ 6x = 180°
∴ x = 30°, 2x = 60° and 3x = 90°

Illustration 5: 

Which of the following pairs of angles are not coterminal?
(A) 330°, − 60°           (B) 405°, − 675°
(C) 1230°, − 930°      (D) 450°, − 630°

Solution:

Here, 405° − (− 675°) = 1080° = 3(360°),
1230° − (− 930°) = 2160° = 6(360°)
and 450° − (− 630°) = 1080° = 3(360°) are a multiple of 360°. Hence, these angles are co-terminal.
Now, 330° − (− 60°) = 390° which is not a multiple of 360°. So, these pair of angles are not co-terminal.

Illustration 6: 

If the arcs of the same length of two circles subtend 75° and 140° at the centre, then the ratio of the radii of the circles is _____.

Solution: 

S1 = S2
∴ r1 × θ1 = r2 × θ2
∴ r1 [75π / 180] = r2 [140 π / 180] ∴ r1 / r2 = 140 / 75
∴ r1 / r2 = 28 / 15

Illustration 7:

The perimeter of a sector of a circle, of area 36π sq.cm., is 28 cm. What is the area of the sector?

Solution:

Area of circle = πr^2 = 36π sq.cm
∴ r = 6 cm
Now, perimeter of sector = 2r + S
But, perimeter is given to be 28 cm.
∴ 28 = 12 + S
∴ S = 16 cm
Area of sector = 1 / 2 × r × S = 1 / 2 × 6 × 16 = 48 sq.cm