In the **circular system** of measurement of angles, the unit of angle
is **radian** (^{c}). A radian, written as 1^{c} is the measure of an angle
subtended at the centre of a circle by an arc equal to the length of its radius
r.

By measuring we can get the value of 1c = 57.27° nearly.

**Note:**

i.
‘^{c}’ used in the
notation of radian is the first letter of the word circular system.

ii.
When no unit is mentioned
with an angle, it is always understood to be in radians.

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**Prove that radian is a constant angle.**

Proof:

Draw a circle with centre O and radius
OA. Take any point B on the circumference such that arc AB is equal to the
length of the radius OA. Join BO. Then ∠BOA = 1^{c}.

Produce AO to C such that CA is a diameter. Let OA = r = arc AB. Arc ABC = half of the circumference = 2Ï€r/2 = Ï€r

Now,

Central angles are proportional to their corresponding arcs.

Since 180° and Ï€ both are constant quantities.
Hence, a radian is a constant angle.

**Relation Among Degree, Grade and Radian Measures**

The angle subtended at the centre of a circle by the total circular arc in radian measure is 2Ï€ radian (^{c}), in degree measure is 360 degree (°)
and in grade measure is 400 grades (^{g}). It follows that,

2Ï€ radian = 360 degree (°) = 400
grades (^{g})

i.e. Ï€ radian = 180 degree (°) = 400
grades (^{g})

Using the above relation, we can
convert degree, grade and radian to each other by using the approximate value of Ï€
as 22/7 or 3.1416.

**Degrees to Radians**

1 degree (°) = Ï€/180 radians (^{c})

∴ x degrees (°) = xÏ€/180 radians (^{c})

**Radians to Degrees**

1 radian (^{c}) = 180/Ï€
degrees (°)

∴ x radians (^{c}) = x × 180/Ï€ degrees (°)

**Grades to Radians**

1 grade (^{g}) = Ï€/200 radian
(^{c})

∴ x grades (^{g}) = x × Ï€/200 radians (^{c})

**Radian to Grades**

1 radian (^{c}) = 200/Ï€ grades
(^{g})

∴ x radians (^{c}) = x × 200/Ï€ grades (^{g})

**Worked Out Examples**

**Example 1:** Express 75° into radian
measures.

**Solution:** Here,

75° = 75 × Ï€/180 radians = 5Ï€/12
radians.

**Example 2:** Express 120g into radian
measures.

**Solution:** Here,

120g = 120 × Ï€/200 radians = 3Ï€/5
radians.

**Example 3:** Express 3Ï€/5 radian (^{c})
into degrees.

**Solution:** Here,

3Ï€/5 radians = (3Ï€/5) × (180/Ï€) = 108
degree (°).

**Example 4:** Express 3Ï€/5 radian (^{c})
into grades.

**Solution:** Here,

3Ï€/5 radians = (3Ï€/5) × (200/Ï€) = 120
grades (^{g}).

**Example 5:** Find the angle between minute
hand and hour hand of a clock at 3:30 pm in radian.

**Solution:** At 3:30 pm, the
distance between minute and hour hand in minute is 12.5 minutes.

Now,

In 60 minutes, minute hand turns 360°

∴ In 1 minute, minute hand
turns 360/60°

∴ In 12.5 minutes, min.
hand turns 12.5 × 360/60°

=
75°

=
75 × Ï€/180 radians

= 5Ï€/12 radians

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