2. 3D - SHAPES UNIT 9 2 1. Polyhedral 1.1. Platonic polyhedral 2. Prisms 3. Pyramids 4. Solids of revolution 5. Area 6. Volume

3. 3D - SHAPES UNIT 9 3 1. POLYHEDRAL

4. 3D - SHAPES UNIT 9 4 Polyhedral Platonic solidsPrismsPyramids Tetrahedron Cube Octahedron Dodecahedron Icosahedron Triangular Quadrangular Pentagonal Hexagonal Triangular Quadrangular Pentagonal Hexagonal

5. 3D - SHAPES UNIT 9 5  A polyhedron is a geometric shape limited by four or more polygons. The limited polygons of the polyhedron are called faces. The common side of two faces is called edge. The common point to three or more edges is called a vertex. FACES EDGES V ERTICES

6. 3D - SHAPES UNIT 9 6 Net Net: It is the surface that is to extend on a plane polyhedron.

7. 3D - SHAPES UNIT 9 7 The polyhedral can be convex or concave. Convex polyherdron, al prolongar cualquiera de sus caras, estas no cortan al poliedro. Concave polyhedron, existe alguna cara que, al prolongarla, corta al poliedro. En los poliedros se cumple la fórmula de Euler:

8. 3D - SHAPES UNIT 9 8  A polyhedron is called regular when these two conditions are fulfilled: Its faces are identical regular polygons. Each vertex of the polyhedron concurs with the same number of faces  There are only five regular polyhedrons: 1.1. Platonic polyhedra Octahedron (8): Formed by 8 equilateral triangles. Tetrahedron (4): Formed by 4 equilateral triangles. Dodecahedron (12): Formed by 12 regular pentagons. Cube or hexahedron (6): Formed by 6 squares. Icosahedron (20): Formed by 20 equilateral triangles.

9. 3D - SHAPES UNIT 9 9 2. PRISMS

10. 3D - SHAPES UNIT 9 10 BASES FACES SIDES  A prism is a polyhedron limited by: Two parallel equal faces that are polygons, called bases. Several parallelograms, called side faces (caras laterales).

11. 3D - SHAPES UNIT 9 11 El desarrollo de un prisma recto está formado por: - Un rectángulo compuesto por sus caras laterales, de altura, la altura del prisma, y ancho, el perímetro de la base. - Los dos polígonos de las bases.

12. 3D - SHAPES UNIT 9 12 Un prisma es un poliedro que tiene dos caras iguales y paralelas entre sí, llamadas bases, y cuyas caras restantes son paralelogramos. Los elementos de un prisma son: Bases o caras básicas: son dos polígonos iguales situados en planos paralelos. Caras laterales: son paralelogramos. Aristas básicas: son los lados de los polígonos de las bases. Aristas laterales: son los lados de las caras laterales que unen las bases.. Vértices: son los puntos donde se cortan las aristas. Altura: es la distancia entre las bases. bases arista arista lateral altura vértice cara lateral

13. 3D - SHAPES UNIT 9 13 Other important elements of a prism are: APOTHEM BASE BASIC EDGE (aristas básicas) SIDE EDGE (aristas laterales) HEIGHT

14. 3D - SHAPES UNIT 9 14 PRISM Oblique Straight IRREGULARREGULAR Classification

15. 3D - SHAPES UNIT 9 15 Nombramos por las bases: TRIANGULAR – HEXAGONAL-… Nombramos por las aristas: RECTO – OBLICUO Nombramos si las bases son polígonos regulares: REGULAR – IRREGULAR Primero miramos los polígonos de las bases. Prisma triangular regular Prisma Pentagonal regular Prisma cuadrangular regular Para nombrar un prisma Regular quadrilateral pyramid Prisma hexagonal oblicuo

16. 3D - SHAPES UNIT 9 16 3. PYRAMIDS

17. 3D - SHAPES UNIT 9 17  A pyramid is a ployhedron with a polygon as its base, and lateral faces are identical isosceles triangles meeting at a, known as the vertex of the pyramid. The distance from the vertex to the base is the height

18. 3D - SHAPES UNIT 9 18 a a´ Side Apotem or face height Side Edge Height of the pyramid Base Apothem Base edge Other important elements of a pyramid are: base

19. 3D - SHAPES UNIT 9 19 PYRAMID OBLIQUE STRAIGHT IRREGULAR REGULAR The base is a regular polygon and the vertex is projected onto the centre of the polygon Classification

20. 3D - SHAPES UNIT 9 20 Pirámide pentagonal recta Pirámide triangular oblicua Pirámide hexagonal recta Nombrando una pirámide:

21. 3D - SHAPES UNIT 9 21 Summarizing:

22. 3D - SHAPES UNIT 9 22 4. SOLIDS OF REVOLUTION

23. 3D - SHAPES UNIT 9 23  Solids of revolution are created by rotating a flat shape around an axis. Exercise. If you rotate these shapes around the indicated axis, which shapes are formed?

24. 3D - SHAPES UNIT 9 24 Cylinder  A cylinder is a solid of revolution generated by a rectangle that rotates around one of its sides. height radius generator Axis of revolution (eje de giro) radius GENERATOR BASE  Elements:

25. 3D - SHAPES UNIT 9 25 Cone  A cone is a solid of revolution generated by a right triangle that rotates around one of its sides.  Elements Generator radius generator Axis of revolution height Axis of revolution Radius BASE

26. 3D - SHAPES UNIT 9 26 Sphere  A sphere is a solid of revolution generated by a circle that rotates around one of its sides. diameter Axis of revolution GENERATOR Center Radius Axis of revolution  Elements

27. 3D - SHAPES UNIT 9 27

28. 3D - SHAPES UNIT 9 28 Ejemplos en el arte y en la vida cotidiana