Convert $40^\circ\, 20'$ into radian measure.

Question

Accepted Answer

> [!TLDR]
> To convert an angle from degrees and minutes to radians, first express it as a pure decimal degree, then multiply by $\frac{\pi}{180}$. For $40^\circ\, 20'$, this gives $\frac{121\pi}{540}$ radians.

The key idea is simple: radian measure is just another way to describe an angle, based on the radius of a circle. One full revolution ($360^\circ$) equals $2\pi$ radians, so the conversion factor is $\frac{\pi}{180}$ radians per degree. But here, the angle isn't given as a clean decimal — it's $40^\circ$ plus $20$ minutes. Minutes are a subdivision of a degree: $1' = \frac{1}{60}^\circ$. So the first step is always to convert the minutes into a decimal part of a degree.

Let's work through it.

1. **Convert minutes to degrees.**  
   Since $1' = \frac{1}{60}^\circ$, we have $20' = 20 \times \frac{1}{60}^\circ = \frac{20}{60}^\circ = \frac{1}{3}^\circ$.

2. **Write the total angle in degrees.**  
   $40^\circ\, 20' = 40^\circ + \frac{1}{3}^\circ = \frac{120}{3}^\circ + \frac{1}{3}^\circ = \frac{121}{3}^\circ$.

3. **Apply the conversion factor to get radians.**  
   Multiply by $\frac{\pi}{180}$:
   $$
   \frac{121}{3}^\circ \times \frac{\pi}{180} = \frac{121\pi}{3 \times 180} = \frac{121\pi}{540}.
   $$

4. **Simplify if possible.**  
   Check if $121$ and $540$ share any common factors. $121 = 11^2$, and $540 = 2^2 \times 3^3 \times 5$. No common factors, so the fraction is already in simplest form.

> [!WARNING]
> A common mistake is to forget that minutes are not decimal degrees. $20'$ is not $0.20^\circ$ — it's $0.333...^\circ$. Always divide minutes by 60.

> [!TIP]
> If you prefer, you can combine steps: $40^\circ\, 20' = \left(40 + \frac{20}{60}\right)^\circ = \frac{121}{3}^\circ$, then multiply by $\frac{\pi}{180}$. The result is $\frac{121\pi}{540}$.

> [!ANSWER]
> The radian measure is $\boxed{\frac{121\pi}{540}}$.

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.Convert $40^\circ\, 20'$ into radian measure.

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.Convert 40∘ 20′40^\circ\, 20'40∘20′ into radian measure.

🎓Unlock everything free for 14 days

Q.Convert $40^\circ\, 20'$ into radian measure.

Unlock everything free for 14 days