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:
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 w l
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 height
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:
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
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), l 2 = r 2 + h 2 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 = r 2 ) Volume (V) of a cone is:
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) = 4 r 2 –Volume radius
9.1 The Rectangular Coordinate System Distance Formula: The distance between 2 points (x 1, y 1 ) and (x 2,y 2 ) is given by the formula: What theorem in geometry does this come from?
9.1 The Rectangular Coordinate System Midpoint Formula: The midpoint M of the line segment joining (x 1, y 1 ) and (x 2,y 2 ) is : Linear Equation: Ax + By = C (standard form)
9.2 Graphs of Linear Equations and Slopes Slope – The slope of a line that contains the points (x 1, y 1 ) and (x 2,y 2 ) is given by: rise run
9.2 Graphs of Linear Equations and Slopes If l 1 is parallel to l 2 then m 1 = m 2 If l 1 is perpendicular to l 2 then: (m 1 and m 2 are negative reciprocals of each other) Horizontal lines are perpendicular to vertical lines
9.3 Preparing to do Analytic Proofs To prove:You need to show: 2 lines are parallelm 1 = m 2, using 2 lines are perpendicular m 1 m 2 = -1 2 line segments are congruentlengths are the same, using A point is a midpoint
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 (x 1, y 1 ) has the equation: y – y 1 = m(x – x 1 )
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)
9.5 Equations of Lines Example (continued): Use the point-slope form with this slope and the point (-4,5) In slope intercept form: