Given,
- External measurements of the wooden box: 10 cm x 8 cm x 6 cm
- Thickness of the wood: 0.5 cm

We need to find the volume of wood used to make the box.

Approach:
- We need to find the volume of the wooden box including the wood used in it and then subtract the volume of the box without the wood to find the volume of wood used.
- Volume of the wooden box including the wood = External volume of the box
- Volume of the box without the wood = Internal volume of the box

Calculation:
1. External volume of the box:
- Length (including both sides of wood) = 10 + 0.5 + 0.5 = 11 cm
- Breadth (including both sides of wood) = 8 + 0.5 + 0.5 = 9 cm
- Height (including both sides of wood) = 6 + 0.5 + 0.5 = 7 cm
- External volume of the box = Length x Breadth x Height = 11 x 9 x 7 = 693 cubic cm

2. Internal volume of the box:
- Length (excluding both sides of wood) = 10 - 0.5 - 0.5 = 9 cm
- Breadth (excluding both sides of wood) = 8 - 0.5 - 0.5 = 7 cm
- Height (excluding both sides of wood) = 6 - 0.5 - 0.5 = 5 cm
- Internal volume of the box = Length x Breadth x Height = 9 x 7 x 5 = 315 cubic cm

3. Volume of wood used:
- Volume of wood used = External volume of the box - Internal volume of the box
- Volume of wood used = 693 - 315 = 378 cubic cm
- But, the thickness of wood is on both sides of length, breadth, and height. So, we need to subtract the volume of empty spaces inside the box due to thickness of wood.
- Volume of empty spaces inside the box = Volume of the box without wood - Internal volume of the box
- Volume of the box without wood = Length x Breadth x Height = 10 x 8 x 6 = 480 cubic cm
- Volume of empty spaces inside the box = 480 - 315 = 165 cubic cm
- Actual volume of wood used = Volume of wood used - Volume of empty spaces inside the box
- Actual volume of wood used = 378 - 165 = 213 cubic cm

Therefore, the volume of wood used to make the box is 165 cubic cm (option B).