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

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 PolyhedraPlatonic 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?