Surface AreaVolumeConclusionSurface Area Volume ConclusionMATH 113Section 10.3: Surface Area and VolumeProf. Jonathan DuncanWalla Walla CollegeWinter Quarter, 2007Surface Area Volume ConclusionOutline1Surface Area2Volume3ConclusionSurface Area Volume ConclusionMeasuring Three Dimensional ShapesIn the last section we looked at two measurements for twodimensional shapes.Two Dimensional ShapesTwo dimensional shapes can be measured in two different ways:1Perimeter (one dimensional)2Area (two dimensional)In the same way, there are two imp ortant types of measurements forthree dimensional shapes.Three Dimensional ShapesThree dimensional shapes have two important measurements:1Surface Area (two dimensional)2Volume (three dimensional)Surface Area Volume ConclusionWhat is Surface Area?Surface area is a term which is mainly used with three dimensionalobjects. It is like the perimeter of a two dimensional object.Surface AreaThe surface area of a three dimensional object is the amount ofexposed area on the figures. For a two dimensional object, thesurface area is simply the area of the object.With many three dime nsional objects we can think of a net for theobject, find the area of the net, and this will be the surface area ofthe object.Surface Area Volume ConclusionSurface Area of Right PrismsWe begin by examining the surface area of right prisms.ExampleFind a general formula for the surface area of a right rectangularprism with a base of length l, width h, and a height of h.Is this formula unique to rectangular prisms?ExampleFind a general formula for the surface area of a right equilateraltriangular prism with base side length s and height h.General FormulaIn general the surface area of a right prism is 2B + ph where B isthe area of the base, h is the height, and p is the perimeter of thebase.Surface Area Volume ConclusionSurface Area of CylindersIn many ways cylinders and prisms are alike. In fact, we can use anet to analyze the surface area.ExampleFind the surface area of a right circular cylinder with radius r andheight h.Surface Area FormulaUsing the formula for prisms, 2B + ph, we can set B = πr2andp = 2πr to get 2πr2+ 2πrh = 2π r (r + h).Surface Area Volume ConclusionSurface Area of Oblique PrismsIn each of the previous examples, we dealt with right objects. Thatis, the sides made right angles with the base. How does surfacearea change in oblique prisms?ExampleCan we develop a surface area formula for an oblique prisms whichis independent of the slant angle?Slant and HeightNote that the length of the lateral faces in an oblique prism willvary depending on the height and the slant angle.Surface Area Volume ConclusionSurface Area of PyramidsBy using nets and the formula for the area of a triangle we candevelop a formula for the surface area of a pyramid.ExampleDevelop a formula for the surface area of a right square pyramidwith a base side length s and a slant height l.General FormulaIn general, a pyramid with an area of base B and perimeter p withslant height l has a surface area B +12pl.Surface Area Volume ConclusionSurface Area of ConesJust as we extended the formula for the surface area of a prism toone for a cylinder, we can extend the surface area for a pyramid toone for a cone.ExampleFind the surface area for a right circular cone with base radius rand slant height h.Surface Area FormulaUsing the formula for a pyramid with a base area of B = π r2andperimeter, or in this case circumference, of 2πr , the surface area isπr2+12(2πr)l = πr (r + l).Surface Area Volume ConclusionSurface Area of a SphereThe final object which we will examine is the s phere.Surface Area of a SphereThe surface area of a sphere of radius r is 4πr2.Radius of a Sphere?What is the radius of a sphere? It is the radius of a “great circle”or largest possible circle taken as a cross-section of the sphere.Unfortunately, to show that this formula works we would need touse calculus. This is beyond the scope of this class.Surface Area Volume ConclusionWhat is Volume?Just as area is measured in “square” units, so volume is measuredin a different sort of unit called a “cubic” unit.VolumeThe volume of an object is the amount of space contained in theobject. It can be thought of as the number of cubes of unit lengthwhich can be fit into the object leaving no empty space and withno overlap.Just as area can be found by multiplying the side length of a polygonby the height of the polygon, we can use multiplication to find thevolume of certain objects.Surface Area Volume ConclusionVolume of Right PrismsThe first, and simplest, object we will examine is a rightrectangular prism.ExampleFind a formula for the volume of a right rectangular prism of baselength l, width w and with height h.ExampleHow many little one by one by one inch cubes could be packedinto a box which is 3 × 5 × 2 inches?Volume FormulaIn general the volume of a right prism is B × h where B is the areaof the base and h is the height.Surface Area Volume ConclusionVolume of Oblique PrismsWhat happens to the volume formula in an oblique prism?ExampleFind a formula for the volume of an oblique rectangular prism ofbase length l, width w , and with height h.ExampleFind a formula for the volume of a right triangular prisms with atriangular base with base length 3 and height 2 having prismheight 5.Surface Area Volume ConclusionVolume of Cylinders and PyramidsHow do these formulas change for cylinders and pyramids?ExampleShow that the volume of a circular cylinder is πr2h where r is theradius of the base and h is the height of the prism.ExampleIn can be shown that three congruent pyramids each with squarebase B can be fit into a cube with sides B. Using this fact, find aformula for the volume of a square pyramid with base area B.Surface Area Volume ConclusionVolume of a SphereAs with surface area, the volume of a sphere is difficult to derive.Volume of a SphereThe volume of a sphere is43πr3.Spheres and CylindersFrom previous work, we know that:The volume of a right circular cylinder of radius r and height2r is 2πr3The volume of a sphere of radius r is43πr3.A sphere of radius r can be inscribed in a right circularcylinder of radius r and height 2r .Therefore, the volume of that sphere is23the volume of thecylinder.Surface Area Volume ConclusionVolume of Irregular ObjectsWhile the formulas above are necessary, their applicability in real live canbe limited as we seldom have perfect prisms, cylinders, pyramids orspheres to work