Cross-Sections of 3D Figures Math Example 4
Follow the full solution, then compare it with the other examples linked below.
Example 4
easyA plane cuts through a sphere. What shape is the cross-section, and when is it the largest possible cross-section?
Solution
- 1 Step 1: Any plane that intersects a sphere produces a circular cross-section (the intersection of a plane and a sphere is always a circle, or a single point if tangent).
- 2 Step 2: The cross-section is largest when the plane passes through the center of the sphere, producing a great circle with radius equal to the sphere's radius .
Answer
The cross-section is always a circle; it is largest (a great circle with radius ) when the plane passes through the center.
Every cross-section of a sphere is a circle because all points on a sphere are equidistant from the center, and the intersection of any plane with this set is a circle. The great circle, formed by a plane through the center, has the maximum possible radius equal to r.
About Cross-Sections of 3D Figures
A cross-section is the flat, two-dimensional shape revealed when a plane cuts through a three-dimensional solid. For example, slicing a cylinder parallel to its base gives a circle, while slicing it at an angle gives an ellipse.
Learn more about Cross-Sections of 3D Figures →More Cross-Sections of 3D Figures Examples
Example 1 easy
A horizontal plane cuts through the middle of a right circular cone (parallel to the base). What 2D
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Example 3 hardA regular hexagonal prism is cut by a plane perpendicular to its bases that passes through two oppos
Example 5 hardA rectangular prism (box) has dimensions [formula] cm [formula] cm [formula] cm. A plane cuts throug