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Understanding Polyhedrons: Rectangular Prisms, Surface Area, and Volume

This section discusses closed spatial figures known as solids, focusing on polyhedrons, particularly rectangular prisms. A polyhedron is defined as a solid composed of polygonal faces, with its edges formed by the intersections of these faces and vertices at their corners. It also covers properties of intersecting, parallel, and skew lines within the context of a rectangular prism. Additionally, it presents formulas for calculating the diagonal, surface area, and volume, providing insights on how to determine these properties for rectangular prisms and related solids.

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Understanding Polyhedrons: Rectangular Prisms, Surface Area, and Volume

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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

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