Measurement Of Angles

Formula : Angle in radians =

Therefore, Angle in radians =.

• Draw a straight line AB.

• Place a dot at B. This dot represents the vertex of the angle.

• Place the centre of the protractor at B and the baseline of the protractor along the arm BA.

• Find 60° on the scale and mark a small dot at the edge of the protractor.

• Join the vertex B to the small dot with a ruler to form the second arm, BC, of the angle.

• Mark the angle with a small arc as shown below.

• Draw a straight line AB.

• Place a dot at B. This dot represents the vertex of the angle.

• Place the centre of the protractor at B and the baseline of the protractor along the arm BA.

• Find 130° on the scale and mark a small dot at the edge of the protractor.

• Join the vertex B to the small dot with a ruler to form the second arm, BC, of the angle.

• Mark the angle with a small arc as shown below.

• Draw a straight line AB.

• Place a dot at B. This dot represents the vertex of the angle.

• Place the centre of the protractor at B and the baseline of the protractor along the arm BA.

• Find 300° on the scale and mark a small dot at the edge of the protractor.

• Join the vertex B to the small dot with a ruler to form the second arm, BC, of the angle.

• Mark the angle with a small arc as shown below.

The given angle is greater than 360°

Adding or subtracting 360° from a particular angle does’nt changes its position.

Therefore, Angle can also be written at as = 430° – 360° = 70°

• Draw a straight line AB.

• Place a dot at B. This dot represents the vertex of the angle.

• Place the centre of the protractor at B and the baseline of the protractor along the arm BA.

• Find 70° on the scale and mark a small dot at the edge of the protractor.

• Join the vertex B to the small dot with a ruler to form the second arm, BC, of the angle.

• Mark the angle with a small arc as shown below.

The given angle is negative

Adding or subtracting 360° from a particular angle does’nt changes its position.

Therefore, Angle can also be written as=-40° + 360° = 320°

• Draw a straight line AB.

• Place a dot at B. This dot represents the vertex of the angle.

• Place the centre of the protractor at B and the baseline of the protractor along the arm BA.

• Find 320° on the scale and mark a small dot at the edge of the protractor.

• Join the vertex B to the small dot with a ruler to form the second arm, BC, of the angle.

• Mark the angle with a small arc as shown below.

Given angle can be completely written in degree as = -220°

-220° = 360° - 220° = 140°

The given angle is negative

Adding or subtracting 360° from a particular angle does’nt changes its position.

Therefore, Angle can also be written as=-310° + 360° = 50°

• Draw a straight line AB.

• Place a dot at B. This dot represents the vertex of the angle.

• Place the centre of the protractor at B and the baseline of the protractor along the arm BA.

• Find 50° on the scale and mark a small dot at the edge of the protractor.

• Join the vertex B to the small dot with a ruler to form the second arm, BC, of the angle.

• Mark the angle with a small arc as shown below.

The given angle is negative

Adding or subtracting 360° from a particular angle does’nt changes its position.

Therefore, Angle can also be written as=-400° + 360° = -40°

The angle is still negative, so we will further add 360° to it.

Therefore, Angle can also be written as=-40° + 360°=320°

• Draw a straight line AB.

• Place a dot at B. This dot represents the vertex of the angle.

• Place the centre of the protractor at B and the baseline of the protractor along the arm BA.

• Find 320° on the scale and mark a small dot at the edge of the protractor.

• Join the vertex B to the small dot with a ruler to form the second arm, BC, of the angle.

• Mark the angle with a small arc as shown below.

Formula : Angle in radians =

Therefore, Angle in radians =

Formula : Angle in radians =

Therefore, Angle in radians =

Formula : Angle in radians =

Therefore, Angle in radians =

Formula : Angle in radians =

Therefore, Angle in radians =

Formula : Angle in radians =

The angle in radians =

Therefore, the total angle =

Therefore, Angle in radians =

Formula : Angle in radians =

Therefore, Angle in radians =

Formula : Angle in radians =

The angle in radians =

Therefore, the total angle =

Therefore, Angle in radians =

(i) Formula : Angle in degrees =

Therefore, Angle in degrees =

(ii) Formula : Angle in degrees =

Therefore, Angle in degrees =

(iii) Formula : Angle in degrees =

The angle in minutes = Decimal of angle in radian x 60.’

The angle in seconds = Decimal of angle in minutes x 60.’’

Therefore, Angle in degrees =

Angle in minutes =

Angle in seconds =

Final angle =

( iv ) Formula : Angle in degrees =

The angle in minutes = Decimal of angle in radian x 60.’

The angle in seconds = Decimal of angle in minutes x 60.’’

Therefore, Angle in degrees =

Angle in minutes =

Angle in seconds =

Final angle =

Let a - d, a, a + d be the three angles of the triangle that form AP.
Given that the greatest angle is double the least.
Now, a + d = 2(a - d)
2a - 2d = a + d
a = 3d ……(1)
Now by angle sum property,
(a - d) + a + (a + d) = 180°
3a = 180°
a = 60° …… (2)
from (1) and (2),
3d = 60°
d = 20°
Now, the angles are,
a - d = 60°- 20° = 40°
a = 60°
a + d = 60° + 20° = 80°.

Therefore the required angles are 40° 60° 80°

The angle in degree =

= 36°

Let, two acute angles are x and y

so,

ATQ, x - y= 36°......(1)

x+ y= 90°......(2)

solving 1 & 2, we get;

⇒ 2x= 126°

⇒ x= 63°

putting the value of x in 2, we get;

⇒ 63°+ y= 90°

⇒ y= 27°

so, Two acute angles are 63° & 27°

Angle in radians =

θ= where θ is central angle, l=length of arc, r=radius

Therefore angle =

Now,

r=

=

Therefore radius is 42 cm

Angle in radians =

θ= where θ is central angle, l=length of arc, r=radius

Therefore angle =

Now,

l = r × θ

=

Therefore the length of the arc is 8.8 cm

Angle in radians =

θ= where θ is central angle, l=length of arc, r=radius

Therefore =

=

l = r × θ

Now, as the length is the same

Therefore,

Therefore the ratio of their radii is 8 : 5

Angle in radians =

θ= where θ is central angle, l=length of arc, r=radius

Now,

= and r = 0.5 x diameter

= radians

θ in degrees =

θ in minutes = 0.5 x 60 = 30’

Therefore angle subtended at the center is 31° 30’

Diameter = 30 cm

Length of chord = 15 cm

Radius = 15 cm [ r = 0.5 x diameter ]

Since the radius is equal to the length of the chord

Hence the formed triangle in the circle is an equilateral triangle.

θ = 60°

We know that l = r × θ

l = = 5 x 3.14 = 15.7

Therefore, the length of the minor arc is 15.7 cm

We know that l = r × θ

Here l = length of arc = 11 cm

R = radius = length of pendulum = 45 cm

We need to find θ

11 = 45 x θ

θ in degree =

For 20 minutes = θ = 4 x 30° = 120°

We know that l = r × θ

l =

Therefore, the length is equal to 88 cm.

Given that Number of revolutions per minute = 180

Then per second, it will be = 180/60 =3

We know that In one complete revolution, the wheel turns at an angle of 2 rad.

Then for 3 complete revolutions, it will take 3 × 2 = 6 radians.

Radius = 1500 m.

Train speed at rate of 66km/hr = 18.33 m/s

Therefore, Distance covered in 1 second = 18.33 m

Distance covered in 10 second = 18.33 × 10 = 183.33m

We know that θ = Distance / radius

θ = 183.33 / 1500

= 0.122 radian

Therefore θ =

θ will be in degrees.

Arc-length can be given by the formula : θ / 360° × 2πr

Hence it is given that 121 cm is the arclength.

⇒ 121 = θ / 360° × 2πr

= 121 = θ / 360° × 2 × 22 / 7 × 180

= 121 = θ / 360° × 360 × 22 / 7

= 121 = θ × 22 / 7

⇒ θ = 121 × 7 / 22

= 38.5°

Hence the angle subtended at the middle is 38.5°

Which can also be written as 38° 30.’

Let the smallest term be x, and the largest term be 2x

Then AP formed= x, ?, ?, 2x

so,

360°= 4/2 [x+ 2x]....[We know that → a+(n-1) d= last term= 2x]

⇒ 180°= 3x

⇒ x= 60°

Now, 60° is least angle.

= 60°= π/180° × 60°

⇒ 60° = π/3 rad