8.1 prisms, area and volume prism – 2 congruent polygons lie in parallel planes –corresponding...

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


• 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).


• 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)


• Volume of a right rectangular prism (box) is given by V = lwh where– l = length– w = width– h = height h

wl


• 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


• Slant height (l) of a pyramid: The altitude of the congruent lateral faces.

height

apothem

Slant height222 hal


• 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.


• 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


• Right circular cone – if the axis which connects the vertex to the center of the base circle is to the plane of the circle.


• 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


• 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


• Regular Polyhedron – is a convex polyhedron whose faces are congruent polygons arranged in such a way that adjacent faces form congruent dihedral angles.

tetrahedron


• Examples of polyhedrons (see book)– Tetrahedron (4 triangles)– Hexahedron (cube – 6 squares)– Octahedron (8 triangles)– Dodecahedron (12 pentagons)


• 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)


• 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


• 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

8.1 prisms, area and volume prism – 2 congruent polygons lie in parallel planes –corresponding...

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