Post on 24-Dec-2015
Surface Area and Volume
Chapter 12
Exploring Solids12.1
California State StandardsLesson goals
8, 9: Solve problems involving the surface area and lateral area of geometric solids and COMMIT TO MEMORY THE NECESSARY FORMULAS.
Identify Polyhedrons
Apply Euler’s Theorem
definitionPolyhedronA 3-dimensional solid figure formed by polygons.
Faces: the polygons that form the polyhedronEdges: a line segment formed by the intersection of two faces.Vertex: the point where 3 or more edges meet.
face
faceedge
vertex
Plural: polyhedra
examplesPolyhedraPrismPyramid
Non-PolyhedraCylinderConeSphere
definition
Regular PolyhedronAll faces are congruent regular polygons
definition
Platonic Solids5 regular, convex polyhedra
Tetrahedron 4 equilateral trianglesHexahedron 6 squaresOctahedron 8 equilateral trianglesDodecahedron 12 regular pentagons Icosahedron 20 equilateral triangles
dodecahedron12 faces
tetrahedron4 faces
hexahedron6 faces(cube)
octahedron8 faces
icosahedron20 faces
definition
Cross-sectionThe intersection of a plane and a solid.
Describe the cross section. The cross section is a square.
definition
Cross-sectionThe intersection of a plane and a solid.
Describe the cross section. The cross section is a circle.
theorem
Euler’s TheoremThe number of faces (F), vertices (V), and
edges (E) of a polyhedron are related by
the formula
2F V E
example
A polyhedron has 18 edges and 12 vertices. How many faces does it have?
exampleFind the number of vertices for a polyhedron with 10 faces made from 4 triangles, 1 square, 4 hexagons, and 1 octagon.
4 triangles 12 sides
1 square 4 sides
4 hexagons 24 sides
1 octagon 8 sides
total 48 sides
Each side is shared by two polygons
48 2 24 edges
10 24Faces Edges
2F V E
10 24 2V
16V
exampleA soccer ball is made of 12 pentagons and 20 hexagons. How many vertices does the soccer ball have?
12 pentagons 60 sides
20 hexagons 120 sides
32 faces 180 sides
180 2 90 edges Each side is shared by two polygons
2F V E
32 90 2V
60V
Definition
back
front
bottomleftside
rightside
leftroof
rightroof
Netthe two-dimensional representation of a three-dimensional figure.
What would the polyhedron look like if laid flat?
exampleWhat would the cylinder look like laid flat?
top
bottom
“label”
Remember: this is called a NET.
Today’s Assignment
p. 723: 6 – 15, 25 – 31, 47 – 49