Volume and Surface AreaThe surface area of a three-dimensional figure is the area that would becovered if the surface were “peeled” off the figure and laid flat. It ismeasured in square units (just like any other area measure).The volume is the measure of cubic units in a three-dimensional figure.A cubic unit is a cube that is one unit long in each of the three dimensions.
BoxThe surface area of a box --- also known as a rectangular parallelepiped ---is the sum of the areas of the six sidesS 2lh 2lw 2hw h w l The volume of a box is the product of the three dimensions. V l w h
Right Circular Cylinder The surface area of a right circular cylinder --- the shape of a soup can or soda can --- is the sum of the areas of the top, the bottom and the side. Th area of the side is the height of the cylinder times the distance around the cylinder. rArea of top = area of bottom= S = 2 r2 + 2 rh r2 hArea of side =2 rh The volume of a cylinder equals the area of the base times the height. V= r 2h
Right Circular ConeThe surface area of a right circular cone --- the shape of a witch’s hat or an icecone --- is the sum of the areas of the base of the cone and the side of the conArea of base = S= r2 + r√r2 + h2 r2 hArea of side = r√r2 + h2 rThe volume of a right circular cone is 1/3 times the area of the base times theV = 1/3 r 2h
SphereTo picture the surface area of a sphere imagine peeling an orange and flattethe peel on a table. The area covered would be the surface area of that sphS = 4 r2 Note this is the area of 4 circles with radius r. rThe formula for the volume of a sphere isV = 4/ 3 r 3
PyramidsA pyramid is a three-dimensional figure with a polygon as its base and isosceletriangles as its sides. For any pyramid if B represents the area of its base andthe height of the pyramid its volume equals 1/3 times the area of the base timethe height.V = 1/3Bh h rectangular pentagonal pyramid pyramid
ApplicationA rectangular office building measures 57 yds. by 36 yds. and is 28 storieshigh. If there are 3.5 yds. per story what is the volume of the building?
ApplicationThe materiel to make an aluminum soda can costs $0.01per square inch. How much does it cost to make a soda can5 in. high and 2.5 in. in diameter? 2.5 in. What is the volume 5 in. of the can?
ApplicationYou work at a local ice cream parlor making waffle cones. If a finished cone6 in. high and has a base diameter of 4 in. what is the surface area of the con(not including the area of the base)? If you fill the cone with soft ice cream topped with a hemisphere of soft ice cream what is the volume of the ice cream?
ApplicationThe Transamerica Tower in San Francisco is a pyramid with a square baseEach side of the base is 52 meters and the height is 260 meters. What is tvolume of the Transamerica Tower?
PolyhedronsA polyhedron is a 3-dimensional figure whose sides or faces are polygoRegular polyhedrons have faces which are all the same regular polygon all equilateral triangles all squares all regular pentagonsThere are only 5 known regular polyhedrons: Tetrahedron --- 4 sides --- all equilateral triangles Hexahedron --- 6 sides --- all squares Octahedron --- 8 sides --- all equilateral triangles Dodecahedron --- 12 sides --- all regular pentagons Icosahedron --- 20 sides --- all equilateral triangles
PrismsA prism is a three-dimensional figure that has two of its faces in parallel planThese faces are congruent polygons. The rest of its faces are parallelogramformed by connecting (drawing a line between) corresponding vertices of thtwo congruent polygons. triangular trapezoidal prism prism
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