1. 8.1 Prisms, Area and Volume
Prism – 2 congruent polygons lie in parallel planes
corresponding sides are parallel.
corresponding vertices are connected
base edges are edges of the polygons
lateral edges are segments connecting corresponding vertices

2. 8.1 Prisms, Area and Volume
Right Prism – prism in which the lateral edges are  to the base edges at their points of intersection.
Oblique Prism – Lateral edges are not perpendicular to the base edges.
Lateral Area (L) – sum of areas of lateral faces (sides).

3. 8.1 Prisms, Area and Volume Lateral area of a right prism: L = hP
h = height (altitude) of the prism
P = perimeter of the base (use perimeter formulas from chapter 7)
Total area of a right prism: T = 2B + L
B = base area of the prism (use area formulas from chapter 7)

4. 8.1 Prisms, Area and Volume
Volume of a right rectangular prism (box) is given by V = lwh where
l = length
w = width
h = height l w h

5. 8.1 Prisms, Area and Volume Volume of a right prism is given by V = Bh
B = area of the base (use area formulas from chapter 7)
h = height (altitude) of the prism

6. 8.2 Pyramids Area, and Volume
Regular Pyramid – pyramid whose base is a regular polygon and whose lateral edges are congruent.
Triangular pyramid: base is a triangle
Square pyramid: base is a square

7. 8.2 Pyramids Area, and Volume
Slant height (l) of a pyramid: The altitude of the congruent lateral faces.
Slant height
height
apothem

8. 8.2 Pyramids Area, and Volume
Lateral area - regular pyramid with slant height = l and perimeter P of the base is: L = ½ lP
Total area (T) of a pyramid with lateral area L and base area B is: T = L + B
Volume (V) of a pyramid having a base area B and an altitude h is:

9. 8.3 Cylinders and Cones
Right circular cylinder: 2 circles in parallel planes are connected at corresponding points. The segment connecting the centers is  to both planes.

10. 8.3 Cylinders and Cones
Lateral area (L) of a right cylinder with altitude of height h and circumference C
Total area (T) - cylinder with base area B
Volume (V) of a cylinder is V = B  h

11. 8.3 Cylinders and Cones
Right circular cone – if the axis which connects the vertex to the center of the base circle is  to the plane of the circle.

12. 8.3 Cylinders and Cones
In a right circular cone with radius r, altitude h, and slant height l (joins vertex to point on the circle), l2 = r2 + h2
Slant height
height
radius

13. 8.3 Cylinders and Cones
Lateral area (L) of a right circular cone is: L = ½ lC = rl where l = slant height
Total area (T) of a cone: T = B + L (B = base circle area = r2)
Volume (V) of a cone is:

14. 8.4 Polyhedrons and Spheres
Polyhedron – is a solid bounded by plane regions. A prism and a pyramid are examples of polyhedrons
Euler’s equation for any polyhedron: V+F = E+2
V - number of vertices
F - number of faces
E - number of edges

15. 8.4 Polyhedrons and Spheres
Regular Polyhedron – is a convex polyhedron whose faces are congruent polygons arranged in such a way that adjacent faces form congruent dihedral angles.
tetrahedron

16. 8.4 Polyhedrons and Spheres
Examples of polyhedrons (see book)
Tetrahedron (4 triangles)
Hexahedron (cube – 6 squares)
Octahedron (8 triangles)
Dodecahedron (12 pentagons)

17. 8.4 Polyhedrons and Spheres
Sphere formulas:
Total surface area (T) = 4r2
Volume
radius

18. 9.1 The Rectangular Coordinate System
Distance Formula: The distance between 2 points (x1, y1) and (x2,y2) is given by the formula: What theorem in geometry does this come from?

19. 9.1 The Rectangular Coordinate System
Midpoint Formula: The midpoint M of the line segment joining (x1, y1) and (x2,y2) is :
Linear Equation: Ax + By = C (standard form)

20. 9.2 Graphs of Linear Equations and Slopes
Slope – The slope of a line that contains the points (x1, y1) and (x2,y2) is given by:
rise
run

21. 9.2 Graphs of Linear Equations and Slopes
If l1 is parallel to l2 then m1 = m2
If l1 is perpendicular to l2 then: (m1 and m2 are negative reciprocals of each other)
Horizontal lines are perpendicular to vertical lines

22. 9.3 Preparing to do Analytic Proofs
To prove:
You need to show:
2 lines are parallel
m1 = m2, using
2 lines are perpendicular
m1 m2 = -1
2 line segments are congruent
lengths are the same, using
A point is a midpoint

23. 9.3 Preparing to do Analytic Proofs
Drawing considerations:
Use variables as coordinates, not (2,3)
Drawing must satisfy conditions of the proof
Make it as simple as possible without losing generality (use zero values, x/y-axis, etc.)
Using the conclusion:
Verify everything in the conclusion
Use the right formula for the proof

24. 9.4 Analytic Proofs
Analytic proof – A proof of a geometric theorem using algebraic formulas such as midpoint, slope, or distance
Analytic proofs
pick a diagram with coordinates that are appropriate.
decide on what formulas needed to reach conclusion.

25. 9.4 Analytic Proofs Triangles to be used for proofs are in: table 9.1
Quadrilaterals to be used for proofs are in: table 9.2.
The diagram for an analytic proof test problem will be given on the test.

26. 9.5 Equations of Lines General (standard) form: Ax + By = C
Slope-intercept form: y = mx + b (where m = slope and b = y-intercept)
Point-slope form: The line with slope m going through point (x1, y1) has the equation: y – y1 = m(x – x1)

27. 9.5 Equations of Lines
Example: Find the equation in slope-intercept form of a line passing through the point (-4,5) and perpendicular to the line 2x+3y=6 (solve for y to get slope of line) (take the negative reciprocal to get the  slope)

28. 9.5 Equations of Lines
Example (continued): Use the point-slope form with this slope and the point (-4,5) In slope intercept form: