8.1 Prisms, Area and Volume

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 l w h

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. Slant height height apothem

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),l2 = r2 + h2 Slant height height radius

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:

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r2 • Volume radius

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?

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)

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

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

9.3 Preparing to do Analytic Proofs

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

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)

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

8.1 Prisms, Area and Volume