Regular hexagon circumscribing a circle:
- The regular hexagon circumscribes the circle such that each vertex of the hexagon lies on the circumference of the circle.
- Since the circle is circumscribed, the distance from the center of the circle to each vertex of the hexagon is equal to the radius of the circle.
- In this case, the radius of the circle is given as 8 cm.

Regular hexagon inscribed in a circle:
- The regular hexagon is inscribed in the circle such that each side of the hexagon is tangent to the circle.
- Since the circle is inscribed, the distance from the center of the circle to each side of the hexagon is equal to the radius of the circle.
- In this case, the radius of the circle is given as 8 cm.

Calculating the area of the hexagons:
- The area of a regular hexagon can be calculated using the formula: Area = (3√3/2) * s^2, where s is the length of a side of the hexagon.
- In the case of the circumscribed hexagon, the length of a side can be calculated using the formula: s = 2 * r, where r is the radius of the circle.
- Therefore, for the circumscribed hexagon, the length of a side is 2 * 8 cm = 16 cm.
- Plugging this value into the area formula, we get: Area of circumscribed hexagon = (3√3/2) * (16)^2 = 192√3 cm^2.

- In the case of the inscribed hexagon, the length of a side is equal to the radius of the circle, which is 8 cm.
- Plugging this value into the area formula, we get: Area of inscribed hexagon = (3√3/2) * (8)^2 = 48√3 cm^2.

Calculating the difference in areas:
- The difference in areas can be calculated by subtracting the area of the inscribed hexagon from the area of the circumscribed hexagon.
- Difference in areas = Area of circumscribed hexagon - Area of inscribed hexagon
= 192√3 cm^2 - 48√3 cm^2
= (192 - 48)√3 cm^2
= 144√3 cm^2.

Final Answer:
The difference in the areas of the regular hexagon circumscribing a circle of radius 8 cm and the regular hexagon inscribed in the same circle is 144√3 cm^2, which is approximately equal to 55.452 cm^2.

Pictorial