CBSE Class 8 Mathematics Visualising Solid Shapes Notes Set 14

Polyhedrons

A polyhedron is a solid bounded by flat polygons called faces. Two faces meet along an edge (a line segment) and edges meet at points called vertices. A sphere, a cylinder and a cone are not polyhedrons because they have curved surfaces.
A polyhedron is convex if the line segment joining any two of its points stays inside or on it.

Regular Polyhedrons

A polyhedron is regular if its faces are congruent regular polygons and the same number of faces meet at every vertex. Only five such solids exist, called Platonic solids: tetrahedron (4 faces, 6 edges, 4 vertices), cube (6, 12, 8), octahedron (8, 12, 6), dodecahedron (12, 30, 20) and icosahedron (20, 30, 12).

Prisms and Pyramids

A prism has two congruent polygons as bases in parallel planes, joined by parallelogram (usually rectangular) faces. On an n-sided base: faces n + 2, vertices 2n, edges 3n.
A pyramid has a polygon as base and triangular faces meeting at one apex. On an n-sided base: faces n + 1, vertices n + 1, edges 2n.
Example: a hexagonal prism has 8 faces, 12 vertices and 18 edges; a hexagonal pyramid has 7 faces, 7 vertices and 12 edges.

Euler's Formula

For every polyhedron, faces + vertices = edges + 2, that is f + v = e + 2.
Example: cube f = 6, v = 8, e = 12 gives 6 + 8 = 14 = 12 + 2. If a polyhedron has 6 faces and 8 vertices, its edges are 6 + 8 - 2 = 12.

Nets

A net is a flat pattern that can be folded to make a solid. Cube: 6 squares. Cylinder: 2 circles and 1 rectangle. Cone: 1 circle and 1 sector. Tetrahedron: 4 triangles. Hexagonal prism: 2 hexagons and 6 rectangles. Square pyramid: 1 square and 4 triangles.
On a dice the numbers on opposite faces add up to 7 (1-6, 2-5, 3-4).