Diamond D’Oveyana & Sylvia SOLIDS Diamond D’Oveyana & Sylvia

Definitions Solids:  the branch of geometry concerned with the properties of three- dimensional geometric figures. Polyhedron: A solid with all flat surfaces that enclose a single region of space Face: Each flat surface Edge: A line segment where the faces of a polygon intersect Vertex: The point where 3 or more edges intersect Base: 2 parallel congruent faces of a polyhedron Surface Area: the sum of the area of all of the faces and side surfaces of a 3D figure Volume: A measure of the amount of  space enclosed by a 3D figure

Regular Polyhedrons- NO CIRCLE Non-Polyhedrons- HAS A CIRCLE Prism- A polyhedron with TWO parallel congruent faces called bases Pyramid- A polyhedron that has a polygonal base and three or more triangular faces that meet at a vertex Cube-A box-shaped solid object that has six identical square faces. Cylinder- A solid with congruent parallel circular bases connected by a curved surface Cone- A solid with a circular base connected by a curved surface to a single vertex Sphere- A set of points in space that are the same distance from a given point. No faces, edges, or vertices Definitions (Cont.)

Polyhedron The 3D shape CANNOT be curved, it has all flat faces Equal sides, equal faces

Cylinder Cone Sphere Torus Non-Polyhedron What certifies a solid to be a non polyhedron is that all or part of the figure is curved. For example: Cylinder Cone Sphere Torus

Platonic Solid Each one is a polyhedron (a solid with flat faces) every face is a regular polygon with the same size and shape.   Example: each face of the cube is a square. They are also convex (no "dents"). They are named after Plato, a famous Greek philosopher and mathematician.

The 5 regular Polyhedra Tetrahedron- 4 triangular faces Hexahedron-6 square faces Octahedron-8 triangular faces Dodecahedron-12 pentagonal faces Icosahedron-20 triangular faces The 5 regular Polyhedra

Side/vertices/edges ratio Platonic Solid # Of Faces # Of Vertices # Of Edges Tetrahedron 4 6 Cube 8 12 Octahedron Dodecahedron 20 30 Icosahedron

Surface Area & Volume The surface area-two- dimensional measurement of the surface of a solid figure. Surface area of a polyhedron is the sum of the areas of each face. Volume-the measure of the amount of space enclosed by a 3D figure Length x Width x Height

l = Slant height r=Radius Formulas Prism Regular Pyramid Cylinder Cone Sphere   T=Ph+2B T=1/2Pl +B T=2πrh+2πr^2 T=πrl +πr^2 T=4πr^2 V=Bh V=1/3Bh V=πr^2h V=1/3πr^2h V=4/3πr^3 T= Total Surface Area P= Perimeter of Base h= Height of Solid V=Volume B=Area of Base l = Slant height   r=Radius

Euler’s Formula for Polyhedrons F-E+V=2 (For any polyhedron that does not intersect itself) The number of faces-the number of edges+ the number of vertices(corner points) always equals 2. For example a cube has 6 faces 8 vertices and 12 edges 6-12+8=2 Every polyhedron has a Euler characteristic