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