Summary of Solid Geometry and Polyhedra
- Positional relationships: intersecting, parallel, and skew (unique to space)
- Line⊥plane: perpendicular to two intersecting lines in the plane
- Euler's theorem: V−E+F=2 (convex polyhedra)
- Regular polyhedra: 5 types (tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron)
- Sphere: volume (4/3)πr³、surface area 4πr²
- Pyramid or cone: volume = 1/3 that of a prism or cylinder
- Three Perpendiculars Theorem: reduce perpendicularity in space to a plane problem
- Height of a regular tetrahedron: h = a√6/3
Comprehensive Practice Problems
- For a polyhedron whose faces are all squares and at whose every vertex three faces meet (a regular hexahedron), verify V, E, and F using Euler's theorem.
- Find the edge length of a regular tetrahedron inscribed in a sphere of radius r.（The circumradius of a regular tetrahedron = a√6/4）
- Find the height and volume of a cone with base radius 3 and slant height 5.
Answers and Explanations
- Regular hexahedron: V=8、E=12、F=6。8−12+6=2 ✓
- Circumradius R = a√6/4 = r → a = 4r/√6 = 4r√6/6 = 2r√6/3
- Height h=√(5²−3²)=√16=4。Volume=(1/3)×π×9×4=12π