Solid Geometry

Cross-sections, nets, and nested solids — the spatial side of 3D shapes.

The idea

Solid geometry studies the shapes of three-dimensional solids — their slices, their surfaces, and how they fit together — rather than their measurement formulas. Because we reason more easily about flat figures than solid ones, each of its main tools turns a solid back into plane geometry.

A cross-section is the flat shape a slicing plane exposes inside a solid. Which shape appears depends on the solid and on the direction of the cut, so one solid can yield several different cross-sections, and naming a cross-section means naming the cut as well.

A net is the solid's surface unfolded flat into one connected piece. The pieces of a net are exactly the solid's faces, so adding their areas gives the solid's surface area.

One solid is inscribed in another when it sits inside and just touches it. The touching forces a relation between the two solids' dimensions, so measuring the outer solid also measures the inner one.

Ways to work on it

• Walkthrough. Cross-sections, nets and surface area, and a sphere inscribed in a cube.
• Practice. Identify a cross-section, or read a box's surface area off its net.
• Hardest. A cone's cross-section, and a square pyramid's surface area via its net.

Not sure where to start? Take the ten-question placement test.