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The problem involves calculating the surface area of a solid created by removing a smaller hemisphere from a larger hemisphere. Here's a breakdown of the steps needed:

### Given:
- **Large hemisphere radius** = 12 mm
- **Small hemisphere radius** = 2 mm
- The formula for the surface area of a sphere is given as \( A = 4\pi r^2 \).

However, we are dealing with hemispheres (half-spheres), so the surface area of a hemisphere is \( 2\pi r^2 \) for the curved part, plus the flat circular base which adds an area of \( \pi r^2 \).

### Total surface area breakdown:
1. **Larger hemisphere surface area**:
   - Curved part of the large hemisphere: \( 2\pi \times 12^2 \)
   - Flat circular base: \( \pi \times 12^2 \)
   - Total surface area of the large hemisphere: \( 3\pi \times 12^2 \)
   
2. **Smaller hemisphere (removed part)**:
   - Curved part of the small hemisphere: \( 2\pi \times 2^2 \)
   - Flat circular base: \( \pi \times 2^2 \)
   - Total surface area of the small hemisphere: \( 3\pi \times 2^2 \)

### Surface area of the final shape:
- We subtract the area of the smaller hemisphere's curved surface, but the flat circular base of the smaller hemisphere leaves a hole, so we do not subtract its base from the overall shape.
  
   \[
   \text{Final surface area} = \left(3\pi \times 12^2\right) - \left(2\pi \times 2^2\right)
   \]

Let's calculate this. I'll do the math for you.The total surface area of the shape, after removing the smaller hemisphere, is approximately **1332 mm²** (to 3 significant figures).

Would you like a further breakdown of the steps, or do you have any questions?

Here are 5 related questions for further exploration:
1. How would the surface area change if the radius of the smaller hemisphere were increased?
2. What would the volume of this shape be if the thickness of the walls were taken into account?
3. How do you compute the surface area for a complex shape involving other geometric solids?
4. How does the surface area formula differ when considering partial spheres (other than hemispheres)?
5. How would this calculation change if the units were given in centimeters instead of millimeters?

**Tip:** Always ensure the consistency of units in problems involving areas or volumes to avoid calculation errors.

The shape below is made from a larger hemisphere with a smaller hemisphere removed from its centre. Work out the surface area of the shape. Give your answer to 3 significant figures. The radii of the hemispheres are 12 mm and 2 mm, respectively.

Learn how to calculate the surface area of a complex 3D shape made by removing a smaller hemisphere from a larger one. This problem involves key geometric concepts, including the surface area formula for spheres and hemispheres. Suitable for students in Grades 9-12.

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