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Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

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Section 12.1 Polyhedron – a 3-D figure whose surfaces are polygons. Face – individual polygon of the polyhedron. Edge – is a segment that is formed by the intersection of two faces. Vertex – is a point where three or more edges intersect.

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Section 12.1 Net – a 2-D pattern that you can fold to form a 3-D figure. Euler’s Formula – the number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula: F + V = E + 2

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CUBE: Net Drawing

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CUBE: 3-Dimensional Faces Edge Vertex

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CYLINDER: Net Drawing

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CYLINDER: 3-Dimensional Faces Edge

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TRIANGULAR PRISM: Net Drawing

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TRIANGULAR PRISM: 3-Dimensional Faces Edge Vertex

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RECTANGULAR PRISM: Net Drawing

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RECTANGULAR PRISM: 3-Dimensional Faces Edge Vertex

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HEXAGONAL PRISM: Net Drawing

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HEXAGONAL PRISM: 3-Dimensional Faces Edge Vertex

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TRIANGULAR PYRAMID: Net Drawing

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TRIANGULAR PYRAMID: 3- Dimensional Slant Height Altitude

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SQUARE PYRAMID: Net Drawing Slant Height

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SQUARE PYRAMID: 3- Dimensional Slant Height

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HEXAGONAL PYRAMID: Net Drawing

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HEXAGONAL PYRAMID: 3- Dimensional Slant Height Altitude

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Chapter 12 – Surface Area and Volume of Solids Section 12.2 – Surface Areas of Prisms and Cylinders

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Section 12.2 Prism – is a polyhedron with exactly two congruent, parallel faces. Bases – two congruent, parallel faces of a prism. Lateral Faces – additional faces of a prism. Altitude – is a perpendicular segment that joins the planes of the bases.

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Section 12.2 Height – the length of the altitude. Right Prism – the lateral faces are rectangles and a lateral edge is the altitude of the prism. Oblique Prism – at least one lateral face is not a rectangle. Lateral Area – is the sum of the area of the lateral faces.

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CUBE: 3-Dimensional BASE LATERAL FACE

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RECTANGULAR PRISM: 3-Dimensional BASE LATERAL FACE

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TRIANGULAR PRISM: 3-Dimensional BASE LATERAL FACE

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HEXAGONAL PRISM: 3-Dimensional BASE LATERAL FACE

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OBLIQUE PRISM: 3-Dimensional BASE LATERAL FACE ALTITUDE

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Section 12.2 Surface Area – the sum of the lateral area and the two bases. Theorem 10-1 – the lateral area of a right prism is the product of the perimeter of the base and the height. L.A. = ph The surface area of a right prism is the sum of the lateral area and the area of the 2 bases. S.A. = L.A. + 2B

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Section 12.2 Cylinder – is a three-dimensional figure with exactly two congruent, parallel faces. Bases – two congruent, parallel faces of a cylinder are circles. Altitude – is a perpendicular segment that joins the planes of the bases.

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CYLINDER: 3-Dimensional BASE

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OBLIQUE CYLINDER: 3- Dimensional BASE ALTITUDE

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Section 12.2 Surface Area – the sum of the lateral area and the two circular bases. Theorem – the lateral area of a right prism is the product of the circumference of the base and the height of the cylinder. L.A. = 2πrh or L.A. = πdh The surface area of a right prism is the sum of the lateral area and the area of the 2 bases. S.A. = L.A. + 2B or S.A. = 2πrh + 2πr 2

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Chapter 12 – Surface Area and Volume of Solids Section 12.3 – Surface Areas and Pyramids and Cones

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Moving from Prisms/Cylinders to Pyramids/Cones

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Section 12.3 Pyramid – is a polyhedron in which one face can be any polygon and the other faces are triangles that meet at a common vertex. Bases – the only face of a pyramid that is not a triangle. Lateral Faces – triangles of pyramid. Vertex of a pyramid – the point where all lateral faces of a pyramid meet.

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Section 12.3 Altitude – is a perpendicular segment from the vertex to the plane of the base. Height – the length of the altitude (h). Regular Pyramid – a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. Slant Height – is the length of the altitude of a lateral face of a pyramid. Lateral Area – is the sum of the area of the congruent lateral faces.

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TRIANGULAR PYRAMID: 3- Dimensional Slant Height Altitude

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SQUARE PYRAMID: 3- Dimensional Slant Height

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HEXAGONAL PYRAMID: 3- Dimensional Slant Height Altitude

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Section 12.3 Surface Area – the sum of the lateral area and the area of the base. Theorem – the lateral area of a regular pyramid is the half the product of the perimeter of the base and the slant height. L.A. = ½ pl The surface area of a regular pyramid is the sum of the lateral area and the area of the base. S.A. = L.A. + B

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Section 12.3 Cone – is a “pointed” like a pyramid, but its base is a circle. Right Cone – the altitude is a perpendicular segment from the vertex to the center of the base. Bases – the only circle on a cone. Vertex of a cone – the only distinctive point on the object.

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Section 12.3 Altitude – is a perpendicular segment from the vertex to the plane of the base. Height – the length of the altitude (h). Slant Height – is the distance from the vertex to a point on the edge of the base. Lateral Area – is ½ the perimeter (circumference) of the base times the slant height.

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CONE: Net Drawing

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CONE: 3-Dimensional

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Section 12.3 Surface Area – the sum of the lateral area and the area of the base. Theorem – the lateral area of a right cone is the half the product of the circumference of the base and the slant height. L.A. = ½ 2 rl or rl The surface area of a right cone is the sum of the lateral area and the area of the base. S.A. = L.A. + B

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Chapter 12 – Surface Area and Volume Section 12.6 – Surface Area and Volumes of Spheres

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Section 12.6 Sphere Set of all points equidistant from a given point. C

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Section 12.6 Surface Area of a Sphere S = 4πr 2 C

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