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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. - PowerPoint PPT Presentation
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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.
• The vertices of the faces are the vertices of the polyhedron.
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
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 F DH 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
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 F Skew Lines: (Some Examples)AB and CG, EH and BF, DC and AE
H G
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
Section 7.1
Surface Area and Volume
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.
Surface Area and Volume
Rectangular Prism• Surface Area• S = 2ℓw + 2wh + 2ℓh
• Volume• V = ℓwh
• ℓ = length• w = width• h = height
Cube• Surface Area• S = 6s²
• Volume • V = s³
• S = Surface Area• V = Volume• s = side (edge)
Section 7.2
Surface Area and Volume of Prisms
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.
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.
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 F S = (18)(12) + 2(20) 4 S = 216 + 40
H 5 G S = 256 un²
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
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 F V = (20)(12) 4 V = 240 un³
H 5 G
Section 7.3
Surface Area and Volume of Pyramids
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.
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.
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.
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 8 B = (9)(12) 10 B = 108 un²
9 S = ½ (10)(42) + 108 12 S = 210 + 108
S = 318 un²
Volume of a Regular Pyramid
• V = ⅓ Bh V = Volume B = Base Area h = height of
pyramid
h = 8 B = (9)(12) 10 8 B = 108 un²
V = ⅓ (108)(8) 9 V = 288 un³ 12
Section 7.4
Surface Area and Volume of Cylinders
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.
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
Surface Area of a Right Cylinder
• S = L +2B or S = 2πrh + 2πr²
S = 2π(4)(9) + 2π4²S = 2π(36) + 2π(16)
9 S = 72π + 32πS = 326.73 un² S = 104π un²
4 (approximate answer) ( exact answer)
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
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)
Section 7.5
Surface Area and Volume of Cones