Surface Area and Volume 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 or Concave Ratio Scale Factor

12.1 Exploring Solids Geometry

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 we look at, state whether it is a polyhedron, if it is state how many faces, edges, and vertices it has.

Types of Solids SOLID POLYHEDRON? FACES EDGES VERTICES 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

F + V = E + 2 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 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?

Problem We have to know how many edges in order to find vertices. Think about each face individually. If we know the shape, we can count the edges What happens when we count the edges for all the faces?

How to fix over counting When counting edges for each face, how many times do we count each edge? How could we fix that? Count all edges for each face ÷ 2 = # of Edges

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

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 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? Having Congruent regular polygons for all its faces How can you find the number of vertices of a polyhedron if you know the number of faces and edges? Use the formula F + V = E + 2 V = E + 2 - F