Cross-Sections & Solids of Revolution

Two ways to move between two and three dimensions.

Slicing: cut a solid with a plane and look at the flat shape you expose. A cylinder cut horizontally gives a circle; cut vertically it gives a rectangle. A cube can give a triangle, square, rectangle, pentagon or even a hexagon depending on the angle.

Rotating: spin a flat shape around a line and it sweeps out a solid.

• A rectangle spun about one side gives a cylinder.
• A right triangle spun about a leg gives a cone.
• A semicircle spun about its diameter gives a sphere.

The axis matters. Spin the same rectangle about its other side and you get a cylinder with different dimensions. Spin it about a line *outside* the shape and you get a ring rather than a solid.

Working out which solid you get, and its radius and height, is most of these problems.

A cross-section is the intersection of a solid with a plane, and its shape depends on the plane's orientation. Cross-sections parallel to a prism's base are congruent to the base; oblique planes produce other figures. A cube admits triangular, quadrilateral, pentagonal and hexagonal cross-sections, the regular hexagon arising from the plane through the midpoints of six edges.

Cross-sections parallel to the base of a pyramid or cone are similar to the base with ratio equal to the fractional distance from the apex, which is exactly the quadratic shrinking that produces the ⅓ factor in the volume formula.

A solid of revolution is generated by rotating a plane region about an axis. The generating region's distance from the axis determines the radius, and its extent along the axis the height. Rotating a region not touching the axis produces a torus or annular solid rather than a simple one.

This is the geometric foundation of the disc and shell methods of integral calculus, where the volume is ∫πr(x)² dx — a continuous version of stacking cross-sections.