Let the angle of the smallest sector be \(a\) and the common difference of the arithmetic progression be \(d\). The angles of the sectors are \(a, a+d, a+2d, a+3d, a+4d, a+5d\).

Given that the largest angle is 4 times the smallest angle, we have:

\(a + 5d = 4a\)

\(5d = 3a\)

The sum of the angles in the circle is \(360^{\circ}\), so:

\(6a + 15d = 360\)

Substitute \(d = \frac{3a}{5}\) into the equation:

\(6a + 15\left(\frac{3a}{5}\right) = 360\)

\(6a + 9a = 360\)

\(15a = 360\)

\(a = 24^{\circ}\)

The arc length of the smallest sector is:

\(\frac{24}{360} \times 2\pi \times 5 = \frac{2\pi}{15} \times 5 = \frac{2\pi}{3}\)

The perimeter of the smallest sector is the sum of the arc length and the two radii:

\(\frac{2\pi}{3} + 2 \times 5 = \frac{2\pi}{3} + 10\)

Approximating \(\pi \approx 3.1416\), the perimeter is:

\(\frac{2 \times 3.1416}{3} + 10 \approx 12.1 \text{ cm}\)