Year+7+3D+Shapes

=Three Dimensional Shapes=

Naming Solids Convex and Non Convex
Convex solids have faces that all point, curve or bulge outwards
 * Convex Solid || Convex Solid || Convex Solid ||
 * [[image:triprism.jpg width="109" height="109"]] || [[image:Sphere.jpg width="109" height="109"]] || [[image:octahedron.jpg]] ||

non-convex solids have some faces that point, curve or cave inwards, having ‘dents’ or ‘holes’.

Polyhedra
Solids can have flat faces and curved faces. A solid whose faces are all flat is called a polyhedron. The plural of polyhedron is polyhedra or polyhedrons The term polyhedron means ‘many faces’ and describes any solid whose faces are polygons. A hexagonal prism is a polyhedron. A cylinder is not because it has a curved face. Polyhedra are named according to the number of faces they have. eg, a tetrahedron has four faces (tetra = 4) a hexahedron has six faces (hexa = 6).

Prisms and Pyramids (right and oblique)
Prisms and pyramids are special types of polyhedra. A prism has the same (uniform) cross-section along its length. Each cross-section is a polygon A cross-section of a solid is a ‘slice’ of the solid, cut across it, parallel to its end faces,rather than along it. The cross-sections of a rectangular prism are congruent (identical) rectangles A prism is either right or oblique, as shown in the diagrams of triangular prisms below An oblique prism stands at an angle or slant

A cross-section of a solid is a ‘slice’ of the solid, cut across it, parallel to its end faces,rather than along it. The cross-sections of a square pyramid are squares but different sizes A pyramid has a pointed top called the apex. The face opposite the apex is a polygon and is called the pyramid’s base. A pyramid’s cross-sections have the same shape as the base but are not the same size. The pyramid on the right has a triangular base, so it is called a triangular pyramid.

Cylinders, Cones and Spheres
Cylinders, cones and spheres are not polyhedra because they have curved faces. Although it has a uniform cross-section, a cylinder is not a prism because its cross-sections are circles, which are not polygons. Although it has an apex, a cone is not a pyramid because its base is a circle, which is not a polygon. Like prisms and pyramids, cylinders and cones can be either right or oblique, as shown below.

A sphere is a perfectly circular solid, the shape of a ball. All of the points on a sphere’s surface are exactly the same distance from the centre of the sphere.

Euler’s Rule for Convex polyhedra
Leonhard Euler (1707–1783) was a famous Swiss mathematician who discovered an interesting rule about polyhedra. He developed a formula relating the number of faces,vertices and edges of a convex polyhedron. An edge is a line of the solid, where two faces meet. A vertex is a corner of the solid, where edges meet.

The prism in the diagram below has six faces, eight vertices and 12 edges Polyhedra are solids that have polygonal faces meeting at edges, and the edges meet at vertices. Let the number of faces, edges and vertices be denoted by F, E and V, respectively. Euler noted that F + V = E + 2 for all polyhedra

Intersecting, parallel and skew edges
The edges are the ‘lines’ of a solid, made where two faces meet Two edges can be related in three different ways. They can be intersecting, parallel or skew. The edges are the ‘lines’ of a solid, made where two faces meet Skew Lines – nonintersecting lines that are not parallel Skew Lines – Two lines are skew if they do not intersect and are not in the same plane

The Platonic solids
A regular polyhedron or Platonic solid is a polyhedron with congruent faces.(faces the same shape and size) Every face is the same regular polygon and all pairs of adjacent faces make equal angles with each other. For example, a cube is a regular polyhedron because every face is a square with sides of the same length. Because a cube has six equal faces, it is also called a regular hexahedron.
 * tetrahedron || Cube ||  ||   ||   ||
 * [[image:tetrahedron.jpg]] || [[image:cube.jpg]] || [[image:octahedron.jpg]] || [[image:dodecahedron.jpg]] || [[image:icosahedron.jpg]] ||