To find the main measurement in radians for $\frac{111\pi}{4}$, you need to express the angle in the standard range for radians, which is between $0$ and $2\pi$.

1. Simplify the given angle: $\frac{111\pi}{4} = \frac{111}{4} \times \pi = 27.75\pi$

2. Find the equivalent angle within $0$ and $2\pi$:

   • Determine how many full rotations (multiples of $2\pi$) are in $27.75\pi$: $27.75\pi \div 2\pi = 13.875$
   • Subtract the integer part (13) to find the fractional part: $13.875 - 13 = 0.875$
   • Multiply the fractional part by $2\pi$ to find the equivalent angle: $0.875 \times 2\pi = 1.75\pi$

So, the main measurement of $\frac{111\pi}{4}$ in radians is $1.75\pi$ radians, or equivalently, $\frac{7\pi}{4}$ radians.

Would you like more details or have any questions?

Here are some related questions:

1. What is the main measurement of $\frac{85\pi}{6}$ in radians?
2. How do you convert an angle from radians to degrees?
3. What is the main measurement of $13.5\pi$ in radians?
4. How do you find the reference angle for $\frac{7\pi}{4}$?
5. What is the difference between radians and degrees?

Tip: Always express angles in the standard range $[0, 2\pi)$ for radians or $[0^\circ, 360^\circ)$ for degrees.