A wheel makes $360$ revolutions in one minute. Through how many radians does it turn in one second?

Question

Accepted Answer

> [!TLDR]
> The wheel's angular speed is found by converting revolutions per minute to radians per second. It turns through $12\pi$ radians in one second.

The key here is understanding what a "revolution" means in angular terms. One full revolution of a wheel corresponds to an angle of $2\pi$ radians — that's the distance around a circle measured in radians, not degrees. So when a wheel spins, every complete turn sweeps out $2\pi$ radians.

The problem gives us a rate: 360 revolutions per minute. But the question asks for the angle turned in *one second*. So we need to convert from minutes to seconds, and from revolutions to radians, in a clean chain.

Let's work through it step by step.

1. **Find the number of revolutions per second.**  
   The wheel makes 360 revolutions in 1 minute. Since 1 minute = 60 seconds, the number of revolutions per second is:
   $$
   \frac{360 \text{ revolutions}}{60 \text{ seconds}} = 6 \text{ revolutions per second}.
   $$

2. **Convert revolutions to radians.**  
   Each revolution is $2\pi$ radians. So in one second, the angle turned in radians is:
   $$
   6 \text{ revolutions} \times 2\pi \text{ radians per revolution} = 12\pi \text{ radians}.
   $$

> [!TIP]
> You can combine both steps into a single calculation:  
> $$
> \frac{360 \text{ rev}}{1 \text{ min}} \times \frac{1 \text{ min}}{60 \text{ s}} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} = \frac{360 \times 2\pi}{60} = 12\pi \text{ rad/s}.
> $$  
> This gives the angular speed directly — but the question asks for the *angle turned* in one second, which is numerically the same as the angular speed in rad/s.

> [!WARNING]
> A common mistake is to forget that one revolution equals $2\pi$ radians, not $\pi$ radians. Another is to mix up minutes and seconds — always check the time units carefully.

> [!ANSWER]
> The wheel turns through $\boxed{12\pi}$ radians in one second.

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

Q.A wheel makes $360$ revolutions in one minute. Through how many radians does it turn in one second?

Concept understanding — Angle Conversion

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

The Intuition: What Is a Radian?

The Key Relationship

How to Convert: Two Simple Steps

Why Bother With Radians?

A Quick Reference Table

The Bottom Line

Q.A wheel makes 360360360 revolutions in one minute. Through how many radians does it turn in one second?

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Q.A wheel makes $360$ revolutions in one minute. Through how many radians does it turn in one second?

Unlock everything free for 14 days