1. Diamond D’Oveyana & Sylvia
SOLIDS
Diamond D’Oveyana & Sylvia

2. 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

3. 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.)

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

5. 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

6. 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.

7. 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

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

9. 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

10. 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

11. 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