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deas531 Chapter 9 link = = =Chapter 7=

Surface Area and Volume
Formulas for surface area of a right rectangular prism with volume v, length l, width w, and height, h are S= 2lw + 2wh + 2lh and V= lwh The surface area, S, and volume, V, of a cube with side s are S= 6s² and V= s³

Example: There are two proposed designs for square candleholders. They have the same volume, but which one has less surface area?

1st candleholder: .4 in x .2 in x 1.0 in 2nd candleholder: .2 in x .5 in x .8 in

1st: 2lw + 2wh + 2lh= .4*.2 + .2*1.0 + .4*1.0= 6.8 in² 2nd: 2lw + 2wh + 2lh= .2*.5 + .5*.8 + .2*.8= 6.6 in²

It's close, but the second candleholder has less surface area.

Site for surface area and volume http://argyll.epsb.ca/jreed/math9/strand3/3107.htm

Surface Area and Volume of Prisms
altitude of a prism - segment that has endpoints in the planes containing the bases and that is perpendicular to both planes height of a prism - same length as an altitude The surface area, S, of a right prism with lateral, L, base area B, perimeter p, and height h is S= hp + 2B Cavalieri's Principle (by me) If two solids have equal heights and cross sections formed by every plane parallel to the bases of both solids have equal areas, then the two solids have equal volumes. Volume of a Prism The volume, V, of a prism with height h and base area B is V= Bh Prisms are named for the shape of their bases.

Example: What is the surface area of a right triangular prism with this net? (also by me) For each base, the area is B= 1/2 (7) * 4= 14 units² The perimeter of each base is p= 9 + 7+ 9= 25 units which means the lateral area is L= hp= 9 * (25) = 225 units² So, SA= 14+ 225= 239 units²

Math lesson for understanding prisms http://www.math.com/school/subject3/lessons/S3U4L2GL.html Another site for prisms http://www.teacherschoice.com.au/Maths_Library/Area%20and%20SA/area_9.htm

Surface Area and Volume of Pyramids
pyramid - polyhedron with a polygon base, and 3 lateral faces lateral face of a pyramid - triangles that share a single vertex vertex of the pyramid - another name for the single vertex that connects all the triangles in a pyramid base edge of a pyramid - the edge that each lateral face has in common with the base lateral edge of a pyramid - the intersection of two lateral faces altitude of a pyramid - the perpendicular segment from the vertex to the base height of a pyramid - length of an altitude regular pyramid - pyramid whose base is a regular polygon and whose lateral faces are all congruent isoceles triangles slant height - length of an altitude of a lateral face of a regular pyramid Pyramids are named for the shape of their base like prisms. The surface area, S, of a regular pyramid with lateral area L, base area B, perimeter of the base p, and slant height l is S= L + B or S= 1/2lp + B The volume, V, of a pyramid with height h and base area B is V= 1/4 Bh

Example: Find the surface area of a right square pyramid with a slant height of 7 and a base edge of 5. S= L + B or S= 1/2lp + B S= 4 (1/2 * 5 * 7) + 5² S= 4 * 17.5 + 25 S= 70 +25 S= 95

Activity for finding the surface area of a pyramid http://www.learnnc.org/lp/media/projects/math/problems/pyramid.pdf

Surface Area and Volume of Cylinders
cylinder - solid that is made up of a circular region and its translated image on a parallel plane lateral surface - curved surface of a cylinder base of a cylinder - the faces formed by the circular region and it's translated image altitude of a cylinder - segment that has endpoints in the planes containing the bases and is perpendicular to both planes height of a cylinder - length of an altitude axis of a cylinder - segment joining the centers of the two bases right cylinder - axis is perpendicular to the bases oblique cylinder - axis is NOT perpendicular to the bases 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 (not copyright because it's currency) Example: A quarter is a right cylinder with a diameter of 24.26 mm and a thickness of 1.75 mm. Imagining the quarter's surface as completely flat, find the surface area of a quarter.

The radius is half of the diameter, so it's 12.13 mm. S= 2πrh + 2πr² S= 2π(12.13) * (1.75) + 2π (12.13)²= 42.455π + 294.2738π 336.7288π about 1,057.3284

Site explaining surface area and volume of a right cylinder [|http://id.mind.net/~zona/mmts/geometrySection/surfaceAreasAndVolumes/areaVolumeCylinder1.html]

Surface Area and Volume of Cones
cone - is a 3 dimensional figure that has a circular base with a curved lateral surface that connects the base that is not in the same plane as the base lateral surface - curved area of a cone vertex - single point in a different plane as the base altitude of a cone - perpendicular segment from the vertex to the plane of the base height of a cone - length of the altitude right cone - altitude of a cone intersects the base at its center oblique cone - altitude of a cone DOES NOT intersect the base of the cone at its center The surface area, S, of a right cone with lateral area L, base area B, radius r, and slant height l is S= L + B or S= πrl + πr² The volume, V, of a cone with radius r, height h, and base area B is V= 1/3Bh or V= 1/3πr²h

Example: Find the surface area of a cone with a slant height, l, of 18 units, and a radius, r, of 6 units. The circumference of the base is c= 2πr= 12π The lateral area is C= 2πl= 32π The part of the circular region occupied by the sector is c/C= 12π/32π= 3/8 The lateral area isπl²= 324π and L= 3/8 * 324π= 121.5π Base area B= πr²= 36π Add the lateral area B + L= 36π + 121.5π= 157.5π= about 494.55

Site for calculating a right cone, with activity http://www.analyzemath.com/Geometry_calculators/surface_volume_cone.html

Surface Area and Volume of Spheres
cone - is a 3 dimensional figure that has a circular base with a curved lateral surface that connects the base that is not in the sphere - a set of all the points in space the same distance from a given point in the center of the sphere The volume, V, of a sphere with radius r is V= 4/3πr³ The surface area, S, of a sphere with radius r is S= 4πr²

Example: The radius of a blown-up balloon is 5 inches. About how many in³ is it holding?

V= 4/3πr³ V= 4/3π (5) ³= 4/3π125 = 4/3 * 125π = 166.66666666625in³ π = about 523.59877559699in³

Site with a sample problem http://www.gomath.com/algebra/sphere.php