Chapter 12 - Surface Area & Volume of Solids

Objectives:

Identify types of solidsCalculate surface area & volume of: Prisms Pyramids Cylinders Cones Spheres

12.1 Exploring Solids

Objectives:Use properties of PolyhedraUse Euler’s Theorem

Polyhedra

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

Polyhedra

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

A vertex of a polyhedron is a point where 3 or more edges meet.

The plural of polyhedron is polyhedra or polyhedrons.

Are these polyhedra?

Types of Solids

Prism

Pyramid

Cone

Cylinder

Sphere

More Terms

A polyhedron is regular if all its faces are congruent regular polygons.

A polygon is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron.

If this segment goes outside the polyhedron, then the polyhedron is nonconvex, or concave.

Cross Sections

Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section.

Did anyone see the Museum of Science exhibition Body Worlds?

Describing cross sections

Platonic Solids

There are 5 regular polyhedra, called Platonic solids after Greek mathematician and philosopher Plato.

Tetrahedron 4 faces Cube 6 faces Octahedron 8 faces Dodecahedron 12 faces Icosahedron 20 faces

How many vertices & edges?

Tetrahedron 4 faces 6 vertices, 6 edgesCube 6 faces 8 vertices, 12 edgesOctahedron 8 faces 6 vertices, 12 edgesDodecahedron 12 faces 20 vertices, 30 edges Icosahedron 20 faces 12 vertices, 30 edges

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

Example 6, p. 722

A soccer ball resembles a polyhedron with 32 faces, 20 regular hexagons and 12 regular pentagons.

How many vertices?Each of the 20 hexagons has 6 sides and each

of the 12 pentagons has 5 sides. Each edge of the soccer ball is shared by 2

polygons.So, the total number of edges is:

E = 1/2 (6*20 + 5*12) = 90

Example 6, p. 722

E = 1/2 (6*20 + 5*12) = 90F + V = E + 232 + V = 90 + 2V = 60