Presentation on theme: "8.1 Prisms, Area and Volume Prism – 2 congruent polygons lie in parallel planes corresponding sides are parallel. corresponding vertices are connected."— Presentation transcript:
1 8.1 Prisms, Area and VolumePrism – 2 congruent polygons lie in parallel planescorresponding sides are parallel.corresponding vertices are connectedbase edges are edges of the polygonslateral edges are segments connecting corresponding vertices
2 8.1 Prisms, Area and VolumeRight 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 prismP = perimeter of the base (use perimeter formulas from chapter 7)Total area of a right prism: T = 2B + LB = base area of the prism (use area formulas from chapter 7)
4 8.1 Prisms, Area and VolumeVolume of a right rectangular prism (box) is given by V = lwh wherel = lengthw = widthh = heightlwh
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 triangleSquare 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 heightheightapothem
8 8.2 Pyramids Area, and Volume Lateral area - regular pyramid with slant height = l and perimeter P of the base is: L = ½ lPTotal area (T) of a pyramid with lateral area L and base area B is: T = L + BVolume (V) of a pyramid having a base area B and an altitude h is:
9 8.3 Cylinders and ConesRight 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 ConesLateral area (L) of a right cylinder with altitude of height h and circumference CTotal area (T) - cylinder with base area BVolume (V) of a cylinder is V = B h
11 8.3 Cylinders and ConesRight 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 ConesIn a right circular cone with radius r, altitude h, and slant height l (joins vertex to point on the circle), l2 = r2 + h2Slant heightheightradius
13 8.3 Cylinders and ConesLateral area (L) of a right circular cone is: L = ½ lC = rl where l = slant heightTotal 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 polyhedronsEuler’s equation for any polyhedron: V+F = E+2V - number of verticesF - number of facesE - 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) = 4r2Volumeradius
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:riserun
21 9.2 Graphs of Linear Equations and Slopes If l1 is parallel to l2 then m1 = m2If 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 parallelm1 = m2, using2 lines are perpendicularm1 m2 = -12 line segments are congruentlengths are the same, usingA 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 proofMake it as simple as possible without losing generality (use zero values, x/y-axis, etc.)Using the conclusion:Verify everything in the conclusionUse the right formula for the proof
24 9.4 Analytic ProofsAnalytic proof – A proof of a geometric theorem using algebraic formulas such as midpoint, slope, or distanceAnalytic proofspick 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 LinesExample: 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 LinesExample (continued): Use the point-slope form with this slope and the point (-4,5) In slope intercept form: