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Section 6.2

Section 6.2. Spatial Relationships. Figures in Space. Closed spatial figures are known as solids . A polyhedron is a closed spatial figure composed of polygons, called the faces of the polyhedron. The intersections of the faces are the edges of the polyhedron.

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Section 6.2

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  1. Section 6.2 Spatial Relationships

  2. Figures in Space • Closed spatial figures are known as solids. • A polyhedron is a closed spatial figure composed of polygons, called the faces of the polyhedron. • The intersections of the faces are the edges of the polyhedron. • The vertices of the faces are the vertices of the polyhedron.

  3. Polyhedrons • Below is a rectangular prism, which is a polyhedron. A B Specific Name of Solid: Rectangular Prism D C Name of Faces: ABCD (Top), EFGH (Bottom), DCGH (Front), E F ABFE (Back), AEHD (Left), H G CBFG (Right) Name of Edges: AB, BC, CD, DA, EF, FG, GH, HE, AE, BF, CG, DH Vertices: A, B, C, D, E, F, G, H

  4. Intersecting, Parallel, and Skew Lines • Below is a rectangular prism, which is a polyhedron. A B Intersecting Lines: AB and BC, BC and CD, D C CD and DA, DA and AB, AE and EF, AE and EH, BF and EF, BF and FG, CG and FG, CG and GH, DH and GH, E FDH and EH, AE and DA, AE and AB, BF and AB, BF and BC, CG and BC H G CG and DC, DH and DC, DH and AD

  5. Intersecting, Parallel, and Skew Lines • Below is a rectangular prism, which is a polyhedron. A B Parallel Lines: AB, DC, EF, and HG; D C AD, BC, EH, and FG; AE, BF, CG, and DH. E FSkew Lines: (Some Examples) AB and CG, EH and BF, DC and AE H G

  6. Formulas in Sect. 6.3 and Sect. 6.4 • Diagonal of a Right Rectangular Prism • diagonal = √(l² + w² + h²). l = length, w = width, h = height • Distance Formula in Three Dimensions • d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] • Midpoint Formula in Three Dimensions • x₁ + x₂ , y₁ + y₂ , z₁ + z₂ 2 2 2

  7. Section 7.1 Surface Area and Volume

  8. Surface Area and Volume • The surface area of an object is the total area of all the exposed surfaces of the object. • The volume of a solid object is the number of nonoverlapping unit cubes that will exactly fill the interior of the figure.

  9. Surface Area and Volume Rectangular Prism Cube Surface Area S = 6s² Volume V = s³ S = Surface Area V = Volume s = side (edge) • Surface Area • S = 2ℓw + 2wh + 2ℓh • Volume • V = ℓwh • ℓ = length • w = width • h = height

  10. Section 7.2 Surface Area and Volume of Prisms

  11. Surface Area of Right Prisms • An altitude of a prism is a segment that has endpoints in the planes containing the bases and that is perpendicular to both planes. • The height of a prism is the length of an altitude.

  12. Surface Area of a Right Prism • S = L + 2B or S = Ph + 2B • S = surface area, L = Lateral Area, • B = Base Area, P = Perimeter of the base, • h = height • The surface area of a prism may be broken down into two parts: the area of the bases and the area of the lateral faces.

  13. Surface Area of a Right Prism • Below is a rectangular prism, which is a polyhedron. A B P = 5 + 4 + 5 + 4 B = (5)(4) D C P = 18 B = 20 12 S = Ph + 2B E FS = (18)(12) + 2(20) 4 S = 216 + 40 H 5 G S = 256 un²

  14. Volume of a Prism • The volume of a solid measures how much space the solid takes or can hold. • The volume, V, of a prism with height, h, and base area, B is: • V = Bh

  15. Surface Area of a Right Prism • Below is a rectangular prism, which is a polyhedron. A B B = (5)(4) D C B = 20 12 V = Bh E FV = (20)(12) 4 V = 240 un³ H 5 G

  16. Section 7.3 Surface Area and Volume of Pyramids

  17. Properties of Pyramids • A pyramid is a polyhedron consisting of one base, which is a polygon, and three or more lateral faces. • The lateral faces are triangles that share a single vertex, called the vertex of the pyramid. • Each lateral face has one edge in common with the base, called a base edge. The intersection of two lateral faces is a lateral edge.

  18. Properties of Pyramids • The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. • The height of a pyramid is the length of its altitude. • A regular pyramid is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. • The length of an altitude of a lateral face of a regular pyramid is called the slant height.

  19. Surface Area of a Regular Pyramid • S = L + B or S = ½ℓp + B. A A is the vertex of the pyramid. B, F, D, and C are the other vertices. Base Edges: BF, FD, DC, CB F Lateral Edges: AB, AC, AD, AF B Base: BFDC Lateral Faces: ∆ABC, ∆ACD, ∆ADF, ∆AFB D The yellow line is the slant height. C The green line is the height of the pyramid.

  20. Surface Area of a Regular Pyramid • S = L + B or S = ½ℓP + B. S = Surface Area L = Lateral Area B = Base Area ℓ = slant height P = 9 + 12 + 9 + 12 ℓ = 10 P = 42 units 8B = (9)(12) 10B = 108 un² 9S = ½ (10)(42) + 108 12 S = 210 + 108 S = 318 un²

  21. Volume of a Regular Pyramid • V = ⅓ Bh V = Volume B = Base Area h = height of pyramid h = 8B = (9)(12) 10 8B = 108 un² V = ⅓ (108)(8) 9V = 288 un³ 12

  22. Section 7.4 Surface Area and Volume of Cylinders

  23. Properties of Cylinders • A cylinder is a solid that consists of a circular region and its translated image on a parallel plane, with a lateral surface connecting the circles. • The bases of a cylinder are circles. • An altitude of a cylinder is a segment that has endpoints in the planes containing the bases and is perpendicular to both bases. • The height of a cylinder is the length of the altitude. • The axis of a cylinder is the segment joining the centers of the two bases. • If the axis of a cylinder is perpendicular to the bases, then the cylinder is a right cylinder. If not, it is an oblique cylinder.

  24. The Surface Area of a Right Cylinder • The surface area, S, of a right cylinder with lateral area L, base area B, radius r, and height h is: • S = L +2B or S = 2πrh + 2πr²

  25. Surface Area of a Right Cylinder • S = L +2B or S = 2πrh + 2πr² S = 2π(4)(9) + 2π4² S = 2π(36) + 2π(16) 9 S = 72π + 32π S = 326.73 un² S = 104π un² 4 (approximate answer) ( exact answer)

  26. Volume of a Cylinder • The volume, V, of a cylinder with radius r, height h, and base area B is: • V = Bh or V = πr²h

  27. Surface Area of a Right Cylinder • V = Bh or V = πr²h V = π(4²)(9) V = π(16)(9) 9 V = π(144) V = 452.39 un³ V = 144π un³ 4 (approximate answer) ( exact answer)

  28. Section 7.5 Surface Area and Volume of Cones

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