If we shade a region in a semi-sphere with radius r, what would be the area of the shaded
region?

To determine the area of a shaded region in a semi-sphere with radius \( r \), we need more details about how the region is defined. For example, is the shaded region a portion of the curved surface of the semi-sphere, or is it some other geometric feature, such as a circular cross-section?

Here are a few common cases for shaded regions on a semi-sphere:

1. **Curved Surface Area of a Semi-Sphere**:  
   The surface area of a full sphere is \( 4\pi r^2 \), so the surface area of a semi-sphere is half of that, or:
   \[
   A_{\text{semi-sphere}} = 2\pi r^2
   \]
   If the entire curved surface of the semi-sphere is shaded, this would be the area.

2. **Circular Cross-Section on the Flat Base**:  
   If the shaded region is a circular cross-section on the flat base of the semi-sphere, the area would simply be the area of a circle with radius \( r \):
   \[
   A_{\text{circle}} = \pi r^2
   \]

3. **Spherical Cap (Partial Surface Area)**:  
   If the shaded region is a spherical cap (a portion of the curved surface cut by a plane), the area depends on the height \( h \) of the cap. The formula for the surface area of a spherical cap is:
   \[
   A_{\text{cap}} = 2\pi r h
   \]
   where \( h \) is the height of the cap from the plane cutting the semi-sphere.

Could you clarify which case applies to the shaded region you're referring to? Depending on the region, the formula will change.

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Would you like more details on any of these cases, or do you have another scenario in mind? Here are 5 related questions to explore further:
1. How do we compute the surface area of a full sphere?
2. What is the area of a spherical sector?
3. Can we calculate the volume of a semi-sphere?
4. How do we find the surface area of a spherical triangle?
5. What is the relationship between the radius and surface area for spheres?

**Tip**: Surface area formulas for spheres usually involve \( \pi \) and powers of the radius \( r \), and can vary based on the geometry of the specific portion considered.

If we shade a region in a semi-sphere with radius r, what would be the area of the shaded region?

This problem explores different cases for finding the area of a shaded region on a semi-sphere with radius r. Learn about surface area formulas for semi-spheres, circular cross-sections, and spherical caps. Suitable for students in Grades 9-12, it covers key geometric concepts and surface area formulas.

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