Open Response A sphere is shown. Describe the shape resulting from a horizontal cross section, a vertical cross section, and an angled cross section.
The Correct Answer and Explanation is:
When a sphere is intersected by different types of cross sections, it creates different two-dimensional shapes.
- Horizontal Cross Section:
A horizontal cross section of a sphere results in a circle. This is because when the plane cuts the sphere parallel to its base, it intersects the sphere’s surface at a constant radius from the center. The resulting shape is a perfect circle, which remains consistent regardless of the height at which the horizontal cut is made, as long as it’s within the sphere’s bounds. - Vertical Cross Section:
A vertical cross section of a sphere also produces a circle. In this case, the plane cuts through the sphere along its vertical axis (i.e., along a plane that passes through the center of the sphere). Since the sphere’s shape is symmetrical, a vertical cut will always create a circular cross section. The diameter of the circle will be equal to the diameter of the sphere. - Angled Cross Section:
When a sphere is intersected by a plane at an angle (neither horizontal nor vertical), the resulting shape will be an ellipse. This occurs because the plane cuts through the sphere at an angle, distorting the circular cross section into an elongated shape. The more tilted the plane, the more elongated the ellipse becomes. In extreme cases, if the angle is steep enough, the ellipse may look almost like a line.
In summary:
- A horizontal cross section creates a circle.
- A vertical cross section also creates a circle.
- An angled cross section creates an ellipse.
