Chapter 12 Surface Area and Volume

Topics We Will Discuss 3-D Shapes (Solids) Surface Area of solids Volume of Solids

Some Vocab We Should Know Equilateral Triangle Polygon Convex Nonconvex Ratio Scale Factor

Skill Review

Geometry 12.1 Exploring Solids

POLYHEDRA A polyhedron is a solid that is bounded by polygons, called faces, that enclose a single region of space. 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 three or more edges meet.

Example of a Polyhedron

Let’s Explore some 3-D Shapes For each shape that you receive. State whether it is a polyhedron, if it is state how many faces, edges, and vertices it has.

Types of Solids SOLIDPOLYHEDRON?FACESEDGESVERTICES PRISM PYRAMID CYLINDER CONE SPHERE CUBE

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

CONVEX POLYHEDRA A polyhedron 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.

CONCAVE POLYHEDRA

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

The 5 Regular Polyhedra Platonic Solids Regular Tetrahedron Cube Regular Octahedron Regular Dodecahedron Regular Icosahedron

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 A solid has 10 faces: 4 triangles, 1 square, 4 hexagons, and 1 octagon. How many vertices does the solid have?

Example A solid has 11 faces: 5 quadrilaterals and 6 pentagons. How many vertices does the solid have?

Example Some quartz crystals are pointed on both ends, and have 14 vertices and 30 edges. If you plan to put a label on one of the faces of a crystal, how many faces do you have to choose from?

Example A paper model of a geodesic dome is composed of 180 triangular faces. How many vertices does it have?

Example Like a soccer ball, a snub dodecahedron has 12 pentagonal faces. The rest of its 92 faces are triangle. How many vertices does the solid have?

Quick Questions What makes a polyhedron a regular polyhedron? How can you find the number of vertices of a polyhedron if you know the number of faces and edges?