Warm-up

• Assemble Platonic Solids

Unit XI: Exploring Surface Area and Volume

•Students will explore nets of three dimensional figures.

•Students will calculate surface area and volume of solid figures, including composite figures.

POLYHEDRA (plural for polyhedron)

• A polyhedron is a solid bounded by polygons, called faces, that enclose a single region of space.

• An edge is a line segment formed by the intersection of two faces.

• A vertex is a point where three or more edges meet.

Am I a Polyhedron?

rectanglesFaces:

Edges:

Vertices:

6

12

8

Am I a Polyhedron?

rectangles and hexagons

Faces:

Edges:

Vertices:

8

18

12

Am I a Polyhedron?hexagon and triangles

Faces:

Edges:

Vertices:

7

12

7

Am I a Polyhedron?

No, it does not have faces that are polygons

Am I a Polyhedron?

No, it does not have faces that are polygons

Am I a Polyhedron?

No, it does not have faces that are polygons

Am I a Polyhedron?

No, it does not have faces that are polygons

Euler’s Theorem

The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula F + V = E + 2.

Use the Euler’s Theorem to find the unknown number.

1. Faces: ____ Vertices: 6 Edges: 12

2. Faces: 5 Vertices: ___ Edges: 9

3. Faces: 20 Vertices: 12 Edges: ___

86

30

Am I a Polyhedron?pentagons

Faces:

Edges:

Vertices:

12

30

€

125• 12( )

€

F +V =E +212 +V =30+212 +V =32V =20

Faces:

Edges:

Vertices:

8 triangles18 squares

48

€

123• 8 +4 • 18( )

€

F +V =E +226 +V =48+226 +V =50V =24

Name the number of faces, edges, and vertices of the polyhedron.

Note: This soccer ball has 32 faces, 20 regular hexagons, and 12 pentagons.Faces: 5

€

edges=124 • 3+1• 4( )

=8

€

F +V =E +25+V =8+25+V =10V =5

Faces: 32

€

edges=1220 • 6 +12 • 5( )

=90

€

F +V =E +232 +V =90+232 +V =92V =60

The Five Platonic Solids - Named after the Greek mathematician

and philosopher Plato

Regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex.

Regular (if all of its faces are congruent) Concave and Convex Polyhedra

concaveregularconvex

irregularconvex

Top View

convex concave

concave concave

Your Turn!!!

• A solid has 14 faces; 6 octagons and 8 triangles. How many vertices does it have?

€

12(8 • 6 +3• 8)

1248 +24( )

1272( )

36 edges

€

F +V =E +214 +V =36+214 +V =38V =24