To solve this problem, let's start by drawing a regular octagon inscribed in a circle of radius 1 cm.

- Draw the circle of radius 1 cm.
- Draw a straight line passing through the center of the circle and dividing it into two equal halves.
- Mark a point on the circle as the fixed vertex.
- Connect the fixed vertex to the center of the circle.
- Draw lines from the fixed vertex to the other vertices of the octagon.

Now, let's analyze the properties of a regular octagon.

- A regular octagon has 8 equal sides and 8 equal angles.
- The distance from the center of the circle to any vertex of the octagon is the radius of the circle, which is 1 cm.

Let's calculate the product of the distance from the fixed vertex to the other seven vertices.

- The distance from the fixed vertex to the center of the circle is 1 cm.
- The distance from the fixed vertex to each of the other seven vertices is equal to the radius of the circle, which is 1 cm.

Therefore, the product of the distance from the fixed vertex to the other seven vertices is 1^7 = 1 cm.

However, we need to be careful with the units in this problem. The question asks for the product of the distances in cm, but the answer choices are given in different units.

To convert the answer choices to cm, we can use the fact that 1 inch is equal to 2.54 cm.

- 4 inches = 4 * 2.54 cm = 10.16 cm
- 8 inches = 8 * 2.54 cm = 20.32 cm
- 12 inches = 12 * 2.54 cm = 30.48 cm
- 16 inches = 16 * 2.54 cm = 40.64 cm

Since the product of the distances is equal to 1 cm, none of the given answer choices match the correct answer.

Therefore, the given answer choices are incorrect, and we cannot determine the correct answer based on the information given.