Solved problems: determining cross-sections of cones and spheres

📚 Understanding Cross-Sections

In geometry, a cross-section is the shape we get when we slice through a 3D object with a plane. The shape of the cross-section depends on the object's shape and the angle of the plane.

📜 Historical Context

The study of conic sections dates back to ancient Greece, with mathematicians like Apollonius of Perga dedicating significant work to understanding these shapes. While the concept of a 'cross-section' as a specific term developed later, the geometric principles were fundamental to their work.

✨ Key Principles for Cones

• 📐 Parallel to the Base: A plane parallel to the base of a cone creates a circular cross-section.
• Ellipse: An ellipse occurs when a plane intersects the cone at an angle that is not parallel to the base, but also doesn't intersect the base.
• ✂️ Tangent to the Side: A plane tangent to the side of the cone yields a triangular cross-section.
• hyperbolas: When the plane is parallel to the axis of the cone, a hyperbola is formed.

🔑 Key Principles for Spheres

• 🔴 Through the Center: A plane passing through the center of a sphere always creates a circular cross-section with the same radius as the sphere. This is called a great circle.
• 🌎 Off-Center: A plane not passing through the center also creates a circular cross-section, but with a smaller radius.

⚗️ Determining Cross-Sections: Step-by-Step

1. 🗺️ Visualize: Imagine the plane slicing through the cone or sphere.
2. 🤔 Consider the Angle: How is the plane oriented relative to the base or axis of symmetry?
3. 📏 Identify Key Points: Where does the plane enter and exit the object?
4. ✍️ Draw the Shape: Based on the above, sketch the resulting shape.

💡 Real-world Examples

• 🍊 Slicing an Orange: When you cut an orange, you are creating cross-sections. Cutting it straight across gives you a circle, while cutting it at an angle gives you something closer to an ellipse.
• 🔦 Flashlight Beam: The beam of a flashlight forms a cone. If you shine it on a wall, the intersection (cross-section) is usually a circle or an ellipse.
• 🌍 Earth's Latitudes: Lines of latitude on a globe are examples of cross-sections of a sphere.

📝 Practice Quiz

1. A cone with a base radius of 5 cm and a height of 12 cm is sliced by a plane parallel to the base at a height of 6 cm from the base. What is the radius of the circular cross-section?
2. A sphere with a radius of 8 cm is sliced by a plane 4 cm from the center. What is the radius of the circular cross-section?
3. Describe the cross-section formed when a cone is sliced by a plane that contains the axis of the cone.

✅ Solutions

1. Using similar triangles: $\frac{r}{6} = \frac{5}{12}$, so $r = 2.5$ cm.
2. Using the Pythagorean theorem: $r = \sqrt{8^2 - 4^2} = \sqrt{48} = 4\sqrt{3}$ cm.
3. The cross-section is an isosceles triangle.

🔑 Conclusion

Understanding the principles behind cross-sections of cones and spheres opens doors to visualizing and analyzing 3D objects more effectively. Keep practicing, and you'll become a master of this geometric concept!