Draw the triangle ABC.

• Draw a line segment $AB = 6 \text{ cm}$.
• Using a compass, place the needle at point A and draw an arc with a radius of $5 \text{ cm}$ (for side AC).
• Place the compass needle at point B and draw an arc with a radius of $4.2 \text{ cm}$ (for side BC).
• The intersection of these two arcs is point C. Connect A to C and B to C to form triangle ABC.

• With the compass needle at A, open it to more than half the length of AB. Draw arcs above and below AB.
• With the same compass opening, place the needle at B and draw arcs that intersect the previous arcs.
• Draw a straight line through the two intersection points. This is the perpendicular bisector of AB.

• Repeat the process from Step 2 for side BC. Place the compass needle at B, open it to more than half the length of BC, and draw arcs.
• With the same opening, place the needle at C and draw arcs intersecting the first set.
• Draw a straight line through these new intersection points. This is the perpendicular bisector of BC.

• The point where the two perpendicular bisectors intersect is the circumcenter of the triangle. Label this point O.

• Place the compass needle at the circumcenter O.
• Adjust the compass opening so that the pencil tip touches any one of the vertices (A, B, or C).
• Draw the circle. This circle will pass through all three vertices of the triangle.

• Use a ruler to measure the distance from the circumcenter O to any of the vertices (OA, OB, or OC). This distance is the radius of the circumcircle.

For reference, the calculated radius using the formula $R = \frac{abc}{4K}$ (where $a, b, c$ are side lengths and $K$ is the area) is approximately:
Let $a=4.2 \text{ cm}$, $b=5 \text{ cm}$, $c=6 \text{ cm}$.
Semi-perimeter $s = \frac{4.2+5+6}{2} = \frac{15.2}{2} = 7.6 \text{ cm}$.
Area $K = \sqrt{7.6(7.6-4.2)(7.6-5)(7.6-6)} = \sqrt{7.6 \times 3.4 \times 2.6 \times 1.6} = \sqrt{107.8976} \approx 10.387 \text{ cm}^2$.
Radius $R = \frac{4.2 \times 5 \times 6}{4 \times 10.387} = \frac{126}{41.548} \approx 3.032 \text{ cm}$.

The radius of the circle should be approximately $\boxed{\text{3.03 cm}}$.
(Note: When performing the physical construction, measure the radius directly from your drawing.)