Convert 6 radians into degree measure.

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Concept understanding — Angle Conversion

Angle Conversion: Why We Need It, and How It Works

Imagine you're measuring the length of a table. You could use centimetres, inches, or feet — all are valid, but the number changes depending on which unit you pick. The same idea applies to angles. An angle is a measure of rotation or opening between two lines, and we have different units to describe that same physical turn.

The two most important units you'll meet are degrees and radians. Degrees are what you likely already know: a full circle is $360^\circ$ . Radians are less familiar but far more natural in mathematics — they're based on the geometry of the circle itself.

The Intuition: What Is a Radian?

Take a circle of radius $r$ . Now walk along the circumference a distance equal to $r$ . The angle you've swept out at the centre is 1 radian.

Note

A radian is the angle that subtends an arc length equal to the radius.

If the arc length $s = r$ , then the angle $\theta = 1$ radian.

So radians directly connect the angle to the arc length:

\theta \text{ (in radians)} = \frac{s}{r}

This is why radians are the "natural" unit — they come from the circle's own geometry, not an arbitrary number like 360.

The Key Relationship

A full circle is $360^\circ$ . In radians, a full circle is the circumference divided by the radius:

\frac{2\pi r}{r} = 2\pi \text{ radians}

So the fundamental conversion is:

360^\circ = 2\pi \text{ radians}

From this, you get the two conversion factors you'll use constantly:

1^\circ = \frac{\pi}{180} \text{ radians}

1 \text{ radian} = \frac{180^\circ}{\pi}

How to Convert: Two Simple Steps

Degrees to radians: Multiply by $\frac{\pi}{180}$ .

Example: Convert $60^\circ$ to radians.

60^\circ \times \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3} \text{ radians}

Radians to degrees: Multiply by $\frac{180^\circ}{\pi}$ .

Example: Convert $\frac{5\pi}{6}$ radians to degrees.

\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = \frac{5 \times 180^\circ}{6} = 150^\circ

Tip

Memorise these common conversions — they appear constantly:

$0^\circ = 0$ , $30^\circ = \frac{\pi}{6}$ , $45^\circ = \frac{\pi}{4}$ , $60^\circ = \frac{\pi}{3}$ , $90^\circ = \frac{\pi}{2}$ , $180^\circ = \pi$ , $270^\circ = \frac{3\pi}{2}$ , $360^\circ = 2\pi$

Why Bother With Radians?

Degrees are fine for everyday use (a right angle is $90^\circ$ , easy). But in calculus, physics, and advanced trigonometry, radians are essential. Here's why:

The derivative of $\sin x$ is $\cos x$ only if $x$ is in radians. In degrees, you'd get an ugly constant factor.
Arc length and area formulas become simple: $s = r\theta$ , $A = \frac{1}{2}r^2\theta$ — these only work with radians.
Many natural phenomena (like oscillations, waves, circular motion) are described cleanly using radians.

Watch out

When you see trigonometric functions in calculus or physics problems, assume the angle is in radians unless explicitly stated otherwise. Using degrees will give wrong answers.

A Quick Reference Table

Degrees	Radians (exact)	Radians (approx)
$0^\circ$	$0$	$0$
$30^\circ$	$\frac{\pi}{6}$	$0.524$
$45^\circ$	$\frac{\pi}{4}$	$0.785$
$60^\circ$	$\frac{\pi}{3}$	$1.047$
$90^\circ$	$\frac{\pi}{2}$	$1.571$
$180^\circ$	$\pi$	$3.142$
$270^\circ$	$\frac{3\pi}{2}$	$4.712$
$360^\circ$	$2\pi$	$6.283$

The Bottom Line

Angle conversion is just changing units — like converting metres to feet. The key is remembering that $360^\circ = 2\pi$ radians. Practice converting a few angles both ways, and it becomes second nature. In exams, always check which unit the problem expects, and if you're using a calculator, make sure it's in the correct mode (DEG or RAD).

Converting between degrees and radians is one of the first skills established in the NCERT Class 11 Mathematics chapter on Trigonometric Functions, and "degree to radian conversion formula and examples" is a heavily searched revision topic for CBSE boards and JEE Main/NEET preparation. Since calculus formulas only hold true in radians, this conversion is a prerequisite that appears repeatedly in "trigonometry important questions" across competitive exams.

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Q.Convert 666 radians into degree measure.