GeomP5 12.1 ­ Solid Figures
Platonic Solids
The Greek philosopher Plato described five solid (3D) shapes that are now known as the Platonic solids. Euclid wrote about them in the Elements, and proved that there are only five of them. Euler came up with a formula that is true of the platonic solids, and in fact true of all polyhedra (many­sided solid figures).
Solid Figures and Polyhedra
A solid figure is a figure that encloses a part of space (or is closed in 3D). A solid figure that is made of only flat surfaces is a polyhedron. The flat surfaces (or polygons) are called faces.
Two faces intersect in a line segment called an edge.
When three faces intersect in a point, that point is a vertex of the polyhedron.
GeomP5 12.1 ­ Solid Figures
Prisms
A prism is a polyhedron that has two parallel faces called bases that are congruent polygons, and the rest of the faces are parallelograms called lateral faces.
All of the edges shared by two lateral faces are called lateral edges, and all lateral edges are parallel.
A prism is classified by the shape of its base.
Pyramids
A pyramid is a polyhedron that has one base that are congruent polygons, and the rest of the faces are triangles called lateral faces.
All of the lateral faces meet at a common point called the vertex.
A pyramid is also classified by the shape of its base.
A tetrahedron is a pyramid with a triangular base.
GeomP5 12.1 ­ Solid Figures
Cylinders and Cones
A cylinder is a solid formed by two bases that are congruent circles, and a curved lateral surface.
Right Cylinder
Oblique Cylinder
A cone is formed by a single base and a curved lateral surface that ends at a point called the vertex.
Right Cone
Oblique Cone
Euler's Formula
In every polyhedron, the number of edges, faces, and vertices follow a formula.
What relationship can you find?