A sphere is a perfectly symmetrical three-dimensional shape in which every point on its surface is equidistant from its center. When we consider the cross sections of a sphere, we can examine both horizontal and vertical cross sections.
Horizontal Cross Section
A horizontal cross section of a sphere is obtained by slicing the sphere parallel to the ground. This slice creates a circular shape. The radius of this circular cross section can vary depending on the height at which the sphere is cut:
- At the center of the sphere, the cross section will be a circle with a radius equal to the radius of the sphere.
- As you move towards the top or bottom of the sphere, the radius of the circular cross section reduces until it reaches zero at the poles (the topmost and bottommost points of the sphere).
Vertical Cross Section
A vertical cross section of a sphere is obtained by slicing the sphere through its center, perpendicular to the horizontal plane. This will create a half-sphere (hemisphere). If you consider cutting it along any vertical plane that passes through the center, the cross section will still be a circle since any vertical slice through the center of a sphere will yield a circular cross section.
- The diameter of this circular cross section will also be equal to the diameter of the sphere at every point along the vertical cut that passes through the center.
Summary
- Horizontal Cross Section: Circular shape that varies in radius depending on the height of the cut.
- Vertical Cross Section: Circular shape (hemisphere) when cut through the center; remains a circle along any vertical plane that passes through the center.
In both cases, the geometric property of circularity is prominent, showcasing the inherent symmetry of the sphere.