1. Surface Area and Volume
Chapter 12
Surface Area and Volume

2. Chapter 12 Objectives Define polyhedron Utilize Euler’s Theorem
Identify cross sections
Identify a prism and cylinder
Identify a pyramid and cone
Calculate the surface area of special polyhedra
Calculate the volume of special polyhedra
Identify a sphere
Calculate the surface area and volume of a sphere
Compare quantities between similar polyhedra

3. Lesson 12.1
Exploring Solids

4. Lesson 12.1 Objectives Define polyhedron Use properties of polyhedra
Utilize Euler’s Theorem

5. Polyhedron A polyhedron is a solid made of polygons.
Remember, polygons are 2-D shapes with line segments for sides.
The polygons form faces, or sides of the solid.
An edge of a polyhedron is the line segment that is formed by the intersection of 2 faces.
Typically the sides of the polygon faces.
A vertex of a polyhedron is a point in which 3 or more edges meet
Typically the corners of the polygon faces.

6. Example of a Polyhedron
Notice that a polyhedron is nothing more than a 3-dimensional shape.
It has some form of a height, a width, and a depth.
Faces
Edges
Vertices
The plural of polyhedron is polyhedra.

7. Example 1 Determine if the following figures are polyhedra.
Explain your reasoning. No
Yes No
There are no faces.
There are no polygons.
All faces are
polygons.
One of the faces
is not a polygon.

8. Regular vs Convex
A polyhedron is regular if ALL of its faces are regular polygons.
Some polyhedron are regular, some are not!
A polyhedron is convex if no line segment connecting two interior points leaves the polyhedron and re- enters.
No dents!
If there are dents, then the polyhedron is concave.

9. Cross Sections
When you take a plane and cut through a solid, the resulting shape of the surface is called the cross section.
When asked to identify a cross section, you need to identify the polygon formed.
The plane acts like a knife blade and cuts through the solid.

10. Theorem 12.1: Euler’s Theorem
The number of faces (F), the number of vertices (V), and the number of edges (E) in a polyhedron are related by
F + V = E + 2
This is commonly used to find one of the variables above.
Typically this is because it is hard to see all the faces, vertices, and edges.
6 + 8 = E + 2
F = 6
14 = E + 2
V = 8
12 = E
E = ???

11. Example 2
Find the number of vertices, faces, and edges each polyhedron has.
F + V = E + 2
F = 5
F = 6
F = 8
F + 8 =
V = 6
V = 6
V = 12
F + 8 = 14
5 + 6 = E + 2
11 = E + 2
6 + 6 = E + 2
= E + 2
F = 6
E = 9
12 = E + 2
20 = E + 2
E = 10
E = 18
You do not have to use the formula to find edges every time. You can use it to find any of the missing quantities as long as you know the other 2, if you use the formula at all!

12. Regular Polyhedra
There are only five regular polyhedra, called Platonic solids.
Named after Greek mathematician and philosopher Plato
They use regular triangles, squares, and regular pentagons to form solids.
Those are the only 3 shapes that can be used.
That is because when their vertices are butted together, their interior angles can add to 360o.

13. Platonic Solids Name Face Shape # Faces Vertices Edges Tetrahedron
Triangle 4 6
Cube
Square 8 12
Octahedron
Dodecahedron
Pentagon 20 30
Icosahedron

14. Homework 12.1
In Class
1-9 p HW
10-55, ev
Due Tomorrow

15. Surface Area of Prisms and Cylinders
Lesson 12.2
Surface Area of Prisms and Cylinders

16. Lesson 12.2 Objectives Identify a prism
Calculate the surface area of a prism
Identify a cylinder
Calculate the surface area of a cylinder
Construct a two-dimensional net for three- dimensional solids

17. Prisms
A prism is a polyhedron with two congruent faces that are parallel to each other.
The congruent faces are called bases.
The bases must be parallel to each other.
The other faces are called lateral faces.
These are always rectangles or parallelograms or squares.
When naming a prism, they are always named by the shape of their bases.
Triangular Prism

18. Parts of a Prism
The edges that connect opposing bases are called lateral edges.
The height of a prism is the perpendicular distance between the bases.
In a right prism, the length of the lateral edge is the height.
A right prism is one that stands up straight with the lateral edges perpendicular to the bases.
In an oblique prism, the height must be drawn in so that it is perpendicular to both bases.
An oblique prism is one that is slanted to one side or the other.
The length of the slanted lateral edge is called the slant height.

19. Examples of Prisms
base
lateral face
slant height
lateral edge

20. Example 3 Answer the following questions:
What kind of figure is the base?
Hexagon
What kind of figure is each lateral face?
Rectangle
How many lateral faces does the figure have? 6
Same number as the sides of the base!
Give the mathematical name of the geometrical solid.
Hexagonal Prism

21. Area
The surface area of a prism is the sum of the areas of all the faces and bases.
The lateral area of a prism is the sum of the areas of the lateral faces.

22. Theorem 12.2: Surface Area of a Prism
The surface area (S) of a right prism can be found using the formula
S = 2B + Ph
B is area of the base
A = ½ aP (any polygon)
A = lw (rectangle)
A = bh (parallelogram)
A = ½ bh (triangle)
A = s2 (square)
P is perimeter of the base
h is height of the prism

23. Example 4 Find the surface area of the following: ? S = 2B + Ph
a2 + b2 = c2
= c2
= c2
169 = c2
c = 13
Example 4
Find the surface area of the following: ?
S = 2B + Ph
S = 2B + Ph
S = 2(5•6) + Ph
S = 2(1/2•12•5) + Ph
S = 2(5•6) + ( )h
S = 2(5•6) + ( )h 13
S = 2(5•6) + ( )(7)
S = 2(5•6) + (30)h
S = 2(30) + (22)(7)
S = 2(30) + (30)(10)
S =
S =
S = 214 m2
S = 360 m2
area

24. Cylinder
A cylinder is a solid with congruent and parallel circles for bases.
height

25. Theorem 12.3: Surface Area of a Cylinder
The surface area (S) of a right cylinder is
S = 2B + Ch
B is area of the base
A =  r2
C is circumference of the base
C = 2 r or
C =  d
h is height of the cylinder

26. Example 5 S = 2B + Ch S = 2r2 + Ch S = 2r2 + 2rh
Find the surface area of the following cylinder.
Round your answer to the nearest hundreth.
S = 2B + Ch
S = 2r2 + Ch
S = 2r2 + 2rh
S = 2(9)2 + 2(9)(7)
S = 2(81) + 2(63)
S = 162 + 126
S = 288
S = =
ft2

27. Nets A net is a two-dimensional drawing of a three-dimensional solid.
If you were to unfold a solid, the net would show what it looks like.
Every solid has a net.
However, there are only certain ways to draw a net for each solid.