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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
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).
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)
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 h
wl
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
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
8.2 Pyramids Area, and Volume
• Slant height (l) of a pyramid: The altitude of the congruent lateral faces.
height
apothem
Slant height222 hal
8.2 Pyramids Area, and Volume
• Lateral area - regular pyramid with slant height = l and perimeter P of the base is:L = ½ lP
• 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: BhV 3
1
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.
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
hrBhV 2
) (22 2rLBLT
rhhCL 2
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.
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
height
radius
Slant height
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:
hrBhV 231
31
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
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
8.4 Polyhedrons and Spheres
• Examples of polyhedrons (see book)– Tetrahedron (4 triangles)– Hexahedron (cube – 6 squares)– Octahedron (8 triangles)– Dodecahedron (12 pentagons)
8.4 Polyhedrons and Spheres
• Sphere formulas:– Total surface area (T) = 4r2
– Volume
radius
334 rV
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?
212
212 yyxxd
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)
2
,2
2121 yyxxM
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:
run
rise
xx
yym
12
12
rise
run
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
121 mm
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
212
212 yyxxd
2
,2
2121 yyxxM
12
12
xx
yym
9.3 Preparing to do Analytic Proofs
• Drawing considerations:1. Use variables as coordinates, not (2,3)
2. Drawing must satisfy conditions of the proof
3. Make it as simple as possible without losing generality (use zero values, x/y-axis, etc.)
• Using the conclusion: 1. Verify everything in the conclusion
2. Use the right formula for the proof
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.
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.
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)
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)
32
32 2
623632
mxy
xyyx
23m
9.5 Equations of Lines
• Example (continued):Use the point-slope form with this slope and the point (-4,5)
In slope intercept form:
11
645
)4(5
23
23
23
23
xy
xxy
xy
23m