To find the radius of the circle when given the area, we can use the formula for the area of a circle:
\[ A = \pi r^2 \]
Given that the area \( A \) is 200 square inches and using \( \pi \approx 3.14 \), we can set up the equation:
\[ 200 = 3.14 r^2 \]
To solve for \( r^2 \), we first divide both sides by \( 3.14 \):
\[ r^2 = \frac{200}{3.14} \]
Calculating the right side:
\[ r^2 \approx \frac{200}{3.14} \approx 63.69 \]
Now, we take the square root of both sides to find \( r \):
\[ r \approx \sqrt{63.69} \approx 7.98 \]
Rounding to the nearest whole number, we find:
\[ r \approx 8 \]
Thus, the approximate measure of the radius is \( \boxed{8} \) inches.