A sphere of radius r inches is sliced by a plane that is k inches from the center. In terms of r and k, what is the radius of the circle of intersection?

To find the radius of the circle formed by the intersection of a sphere with a plane, we can use the following geometric principles.

1. **Understanding the problem:**
   - We have a sphere of radius \( r \).
   - A plane slices the sphere \( k \) inches away from the center of the sphere.

2. **Visualize the Geometry:**
   - When a plane intersects a sphere, the intersection is a circle.
   - The distance from the center of the sphere to the plane is \( k \).
   - We need to find the radius of this circle of intersection.

3. **Using the Pythagorean Theorem:**
   - Let's denote the radius of the circle of intersection as \( R \).
   - Consider a right triangle formed by:
     - The radius \( r \) of the sphere.
     - The distance \( k \) from the center of the sphere to the plane.
     - The radius \( R \) of the circle of intersection.

   The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (in this case, \( r \)) is equal to the sum of the squares of the other two sides (\( R \) and \( k \)).

   \[
   r^2 = R^2 + k^2
   \]

4. **Solve for \( R \):**
   \[
   R^2 = r^2 - k^2
   \]
   \[
   R = \sqrt{r^2 - k^2}
   \]

Therefore, the radius \( R \) of the circle of intersection is given by:

\[
R = \sqrt{r^2 - k^2}
\]

### Conclusion
The radius of the circle of intersection in terms of \( r \) and \( k \) is \( \sqrt{r^2 - k^2} \).

Would you like more details or have any questions? Here are some related questions to explore further:

1. How do you derive the equation of a sphere in three dimensions?
2. What is the volume of the sphere in terms of \( r \)?
3. How do you find the equation of a circle in a plane?
4. What is the surface area of a sphere?
5. How does the distance of the plane from the center affect the radius of the intersection circle?
6. How do you calculate the area of the circle of intersection?
7. Can you explain the Pythagorean theorem in the context of this problem?
8. What happens to the radius of the circle of intersection if \( k \) is greater than \( r \)?

**Tip:** Visualizing the problem using diagrams can greatly aid in understanding geometric relationships.

Learn how to find the radius of the circle formed by the intersection of a sphere and a plane, given the sphere's radius and the distance of the plane from the center. This problem uses the Pythagorean theorem to derive the formula sqrt(r^2 - k^2). Suitable for students in Grades 9-12.

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