Which regular polygon has an angle of:
1. 60 degrees
2. 108 degrees
3. 90 degrees
4. 120 degrees
- triangle
- pentagon
- square
- hexagon
What is a polyhedra?
A 3-DIMENSIONAL SOLID, BOUNDED BY POLYGONS
What conditions must be met for a convex polyhedron to be deemed regular?
- all bounding polygons are congruent regular polygons
- each vertex is adjacent to the same number of bounding polygons
What is another name for a regular convex polyhedron?
A PLATONIC SOLID
How many platonic solids are there? List them.
- tetrahedron
- cube
- octahedron
- dodecahedron
- icosahedron
What are the 3 parts of polyhedra?
- face
- vertex
- edge
What is the Euler Characteristic formula?
F - E + V = EC
*** if
- F is the number of Faces,
- E is the number of Edges,
- and V is the number of vertices
What is the Euler Characteristic of all convex polyhedra?
2
How many faces does the tetrahedron have?
4
How many faces does the cube have?
6
How many faces does the octahedron have?
8
How many faces does the dodecahedron have?
12
How many faces does the icosahedron have?
20
What conditions must be met for a convex polyhedron to be deemed semiregular?
- all bounding polygons are regular polygons with edges the same length
- and if each vertex is adjacent to the same number of bounding polygons
- and there exists a fixed cyclic order of the types of polygons around all the vertices
What kinds of semiregular polyhedra are there? How are they defined?
- prism:
- identical ends (parallel top and bottom);
- squares in the side band;
- cross sections are identical - antiprism:
- identical ends (parallel top and bottom);
- equilateral triangles in the side band;
- cross sections are different from one another
What is an Archimedean Solid? How many are there?
- semiregular polyhedra BUT NOT prism or antiprism
- 13 (that are semiregular polyhedra)
T/F: Not all prisms and antiprisms are semiregular polyhedra
FALSE
- all prisms and antiprisms are semireg. polyhedra
Exercise: A rhombicuboctahedron is an archimedean solid. It has 24 vertices, each which meets 3 squares and one triangle. How many faces does it have? How many edges?
Exercise: A icosododecahedron is an archimedean solid that has 12 pentagon faces and 20 triangle faces. How many vertices does it have? How many edges?
Why are there only 5 Platonic Solids?
If the internal angles that meet at a given vertex add up to more than 360 degrees, the shape will flatten (become 2D)
Why is a tetrahedron a platonic solid?
It’s the meeting of 3 reg. triangles —- internal angles equal 180 degrees
*LESS THAN 360 DEGREES
Why is an octahedron a platonic solid?
It’s the meeting of 4 reg. triangles —- internal angles equal 240 degrees
*LESS THAN 360 DEGREES
Why is an icosahedron a platonic solid?
It’s the meeting of 5 reg. triangles —- internal angles equal 300 degrees
*LESS THAN 360 DEGREES
