Visualizing solid shapes - by Elwin binu
- 2. 2D and 3D figures 1. Two Dimensional Figures: These are plane figures which just have measurements like length and breadth. Example: Rectangle, Square, etc. 2. Three Dimensional Figures: These are solid figures which have measurements like length, breadth, height. Example: Cube, cuboid , etc.
- 4. Two types of 3D objects • Non-polyhedrons 3D objects which have a curved surface. Eg. Cylinder, Cone, Sphere • Polyhedrons 3D objects which are made of flat surfaces (polygons). In geometry, a polyhedron is a three dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron. Eg. Everything else…. Like,
- 5. Types of polyhedrons • All polyhedrons are classified based on the characteristics of the polygons associated with them.
- 6. Regular and Irregular polyhedrons Based on regularity of the polyhedron’s polygon. Based on number of faces meeting at each vertex being equal or not.
- 7. Some examples All angles are equal and all sides in the polygon are equal
- 8. Convex and non-convex polyhedrons Convex polyhedrons Concave polyhedrons Polyhedrons Check if :- a) All the interior angles polyhedron’s polygons are < 180°. b) All vertices point outward. (Outward=Convex , Inward=Concave)
- 10. Regular and Irregular polyhedrons REGULAR • Equal sides and angles in its polygon • Same number of faces converge at a vertex IRREGULAR • Unequal faces and sides in its polygon • Not the same number of faces converge at a vertex
- 11. Convex and Non-Convex polyhedrons Convex • All interior angles are less than 180 degrees • Two points on the surface of the polygon can always be connected without protruding the polygon • All the vertices of the polygon will point outwards, away from the interior of the shape. Think of it as a 'bulging' polygon. Non-Convex • At least one interior angle is greater than 180 degrees • Two points on the surface of the polygon cannot always be connected without protruding the polygon • At least one of its vertices will be point inward, towards the polygon.
- 12. The 5 regular, convex polyhedrons • There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. Mr. Plato
- 13. Prisms and Pyramids • Prisms and Pyramids are two important members of the polyhedron family. • Prisms have 1) identical parallel ends (congruent base and top-Rectangles) 2) all faces flat • Pyramids have one base and all other faces triangles • Prisms have two bases -- pyramids only have one. • Both are named based on their bases.
- 14. • Prisms have two bases, which can be any polygon. All the other faces(lateral faces) are rectangles. • Named based on type of base.
- 15. • Pyramids have only one base, which can be of any polygon. All the other faces are triangles. • Named after type of base.
- 16. Euler’s formula • Euler’s formula states that the sum of number of faces and vertices = the number of edges + 2. F + V = E - 2 • So, F + V – E = 2
- 17. Let’s try and prove his theory
- 21. Euler’s formula • Euler’s formula states that the sum of number of faces and vertices = the number of edges + 2. F + V = E - 2 • So, F + V – E = 2