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Surface Area & Volume. Vocabulary. Polyhedron – a three-dimensional figure whose surfaces are polygons Face – each of the polygons of the polyhedron Edge – a segment that is formed by the intersection of two faces Vertex – a point where three or more edges intersect. Vocabulary.

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## Surface Area & Volume

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**Vocabulary**• Polyhedron – a three-dimensional figure whose surfaces are polygons • Face – each of the polygons of the polyhedron • Edge – a segment that is formed by the intersection of two faces • Vertex – a point where three or more edges intersect**Vocabulary**• Polyhedron – a 3-dimensional (3-D) solid with faces and edges**Prisms**• A prism is a polyhedron with exactly two congruent, parallel faces, called bases. • Other faces are lateral faces (also congruent) • An altitude of a prism is a perpendicular segment that joins the planes of the bases. • The heighth of the prism is the length of an altitude.**Prisms**• Prism – are named by the shape of their bases. • This one is a _________________ prism.**Prisms**• Prism – are named by the shape of their bases. • This one is a _________________ prism.**Prisms**• Prisms – are named by the shape of their bases. • This one is a _________________ prism.**Prisms**• Have bases and faces that are polygons – therefore, these sides have all of the properties of polygons • One of those properties is area.**Prisms**• Net – is what we would have if we took a polyhedron an unfolded it. • If we made a cube into a net, here is what we would have:**Prisms**• If we made a triangular prism into a net, here is what we would have:**Surface Area of Prisms**• Surface area – is just that, the area of the surface of the entire polyhedron. • How many surfaces does this have?**Surface Area of Prisms**• How might we go about finding the surface area of this square polyhedron?**Surface Area of Prisms**• Surface area = area of all of the lateral faces + area of the two bases**Surface Area of Prisms**• The lateral area of a prism is the sum of the areas of the lateral faces. • The surface area is the sum of the lateral area and the area of the two bases.**Surface Area of Prisms**• The lateral area of a right prism is the product of the perimeter of the base and the height. LA = ph • The surface area of a right prism is the sum of the lateral area and the areas of the two bases SA = LA + 2B**Surface Areas of Prisms**• Use a net to find the surface area. 8 ft 7 ft 6 ft**Surface Areas of Prisms**• Use a formula to find the lateral area and surface area. • L.A.=ph=24*7=168 ft2 • S.A.= L.A.+2 B= 216 ft2 8 ft 7 ft 6 ft**Homework**• Handout on Surface Area of Prisms**Surface Areas of Cylinders**• A cylinder has two congruentparallel bases. However, the bases of a cylinder are a circle. • An altitude of a cylinder is a perpendicular segment that joins the planes of the bases. • The heighth of a cylinder is the length of an altitude.**Surface Areas of Cylinders**Right cylinder Oblique cylinder**Surface Areas of Cylinders**• The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder. LA = 2πrh or LA = πdh • The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases. SA = LA + 2πr2**Surface Areas of Cylinders**• The radius of the base of a cylinder is 16 in., and its height is 4 in. Find its surface area in terms of π. S.A.= 2πrh + 2πr2= 640 π**Homework**• Handout on surface area of cylinders**Volumes ofPrisms and Cylinders**• Volume is the space that a figure occupies. It is measured in cubic units. • Volume of a prism V = Bh B - area of the base For a rectangular prism, V = (l * w) * h**Volumes ofPrisms and Cylinders**• Volume of a cylinder V = Bh B - area of the base • In a cylinder, B= πr2, so V = πr2h.**Volumes ofPrisms and Cylinders**• Find the volume of the figure to the nearest whole number. V = Bh V= (½ *3*5)*6 V= 45 in.3 6 in. 5 in. 3 in.**Volumes ofPrisms and Cylinders**• Find the area of the figure to the nearest whole number. • V = πr2h • V = π(2)25= 63m3 5 m 2 m**Homework**• Handouts on the volume of prisms and cylinders**Surface Area**• Review: • Prism – a 3-D polyhedron with 2 bases and a height that connects them • Cylinder – a 3-D polyhedron with circles for bases • Surface Area – the area of all of the sides of a polyhedron added together • Net – a polyhedron unfolded into a flat, 2-dimensional figure**Surface Area of Pyramids**• A pyramid: • Is a polyhedron • Has only 1 base that is a polygon • Has sides (lateral faces) that are triangles • Has sides that meet at a vertex • Has a height and a slant height • We name pyramids by the shape of the base.**Surface Areas ofPyramids**• A regularpyramid is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. • The slant height l is the length of the altitude of a lateral face of the pyramid.**Surface Areas ofPyramids**• The lateral area of a regular pyramid is the sum of the areas of the congruent lateral faces. LA = ½ pl p – perimeter of the base l – slant height • The surface area of a regular pyramid is the sum of the lateral area and the area of the base. SA = LA + B**Surface Areas ofPyramids**• Find the slant height of a square pyramid with base edges 12 cm and altitude 8 cm. • First, draw and label the figure. • l= 10 cm**Surface Areas ofPyramids**• Find the lateral area and surface area of the regular square pyramid below. LA = ½ pl = ½ (4*4 sides) 7 = 56 in.2 SA = LA + B = 56 + (4*4) = 72 in.2**Homework**• Handout on surface area of pyramids**Surface Areas ofCones**• A cone is like a pyramid, but its base is a circle. • The slant height is the distance from the vertex to a point on the edge of the base.**Surface Areas ofCones**• The lateral area of a right cone is half the product of the circumference of the base and the slant height. LA = ½ (2πr)l = πrl • The surface area of a right cone is the sum of the lateral area and the area of the base. SA = LA + πr2**Surface Areas ofCones**• Find the lateral and surface area of a cone with radius 8 cm and slant height 17 cm. Leave your answer in terms of π. • LA = πrl • LA= 136π cm2 • SA = LA + πr2 • SA= 136π + 64π= 200πcm2**Surface Area of Cones**• Find the Surface Area of a cone with a base with radius 10ft and a slant height of 12ft: • LA = πrl • LA= 120π ft2 • SA = LA + πr2 • SA= 120π+ 100π= 220πft2**Homework**• Handout on Surface Area of Cones**Volumes ofPyramids and Cones**• Volume of a pyramid V = 1/3 Bh • In a cone, B= πr2, so V = 1/3 πr2h.**Volumes ofPyramids and Cones**• Find the volume of the figure. • V = 1/3 Bh • V= 1/3 (6*6) *5 • V= 60 ft3**Volumes ofPyramids and Cones**• Find the volume of a cone with diameter 3 m and height 4 m. • V = 1/3 πr2h • V= 1/3 π(3/2)24 • V= 3π m3**Volumes ofPyramids and Cones**• Find the volume of a square pyramid with base edges 24 in. long and slant height 15 in. • V = 1/3 Bh • First find h. • 122+h2=152 • h=9 • V= 1/3 (24*24)*9= 1728 in.3**Homework**• Handouts on Volume of Pyramids and Cones**Surface Areas and Volumesof Spheres**• A sphere is the set of all points in space equidistant from a given point called the center.**Spheres**• Half of a sphere is called a hemisphere. • How do you think the volume of a hemisphere would compare to that of a sphere?**Spheres**• What is the relationship between a sphere and a cylinder? • Same radius; volume of a sphere is less**Surface Area of a Sphere**• A sphere is a 3-Dimensional object and a circle is only 2-D (flat). • What do we think we can say about the surface areas of the two in comparison to each other?**Surface Area of a Sphere**• If the surface area of a sphere is more than the area of a circle, then the formula must give us a value greater than that of a circle. • Area of a Circle = π r2 • Surface area of a Sphere = 4(π r2)

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