Chapter Ten. Volumes and Surface Areas of Simple Solids

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1 Capter Ten Voumes and Surface Areas of Simpe Soids We ook at o to cacuate te areas and perimeters of many geometric figures in te ast capter. As you get farter advanced in mat, you find cases ere you need to find te voume of a soid object or te tota surface area. Because tis is a beginner s book, e re going to cover ony a fe of te simper soid objects. Te coor iustrations i ep you to visuaize eac probem. Voume is expressed in cubic units. So, for instance, if you ave an object measured in inces, te voume oud be given in cubic inces. For an object measured in centimeters, te voume oud be in cubic centimeters. Voume and Surface Area of a Rectanguar Prism Te voume of a rectanguar prism is te product of te engt, idt, and eigt. Tis is one of te most common voume cacuations tat you i do. For exampe, tis oud te you o muc space tere is in a room of your ouse or o muc ater an aquarium ods. Voume = engt x idt x eigt Voume = x x Voume = Te surface area of a rectanguar prism is aso easy to cacuate. First, you i notice tat for any given rectanguar prism, tere are tree sets of faces. Te top and bottom face are te same, te front and backfaces are te same, and te eft and rigt faces are te same. Doctor Wat s Mat Magic Book One

2 eft side front top bottom back rigt side I ve son te faces of te rectanguar prism in expoded vie to te eft to simpify our discussion. Te surface area of te rectanguar prism is te sum of te areas of a six faces. Here are te area cacuations for te six faces. Area of front = x Area of back = x Area of top = x Area of bottom = x Area of eft side = x Area of rigt side = x So te tota surface area can be cacuated as foos: Tota Area = 2 ( x ) + 2 ( x ) + 2 ( x ) Let s sove a practica probem using at e ave just earned. Suppose tat you need a storage space tat as at east 1200 cubic feet for a project. Your friend offers te use of a room tat is 12ft ide x 14ft ong x 8 ft ig. Is te room arge enoug? You cacuate te voume using: V = x x V = 14 x 12 x 8 = 1344 cubic feet. So te room is arge enoug to use. Doctor Wat s Mat Magic Book One

3 Let s say tat you decide to paint te four as and te ceiing prior to putting your stuff in te room. Wen you ook at paint at te store, te abe says tat a gaon i cover 300 square feet. Ho many gaons of paint soud you buy to put one coat on eac a and te ceiing? Te tota surface area of te room is cacuated using: Tota Area = 2 ( x ) + 2 ( x ) + 2 ( x ) Area of foor = x Paintabe Area = 2 ( x ) + ( x ) + 2 ( x ) Paintabe Area = 2 ( 14 x 8) + ( 14 x 12) + 2 ( 12 x 8) Paintabe Area = = 584 So it ooks ike you better buy to gaons. Voume and Surface Area of a Cube A cube is a specia case of a rectanguar prism. It as six identica faces. So te engt equas te idt equas te eigt. Tis makes te cacuation of voume and surface area quite easy. S S Voume = S x S x S = S 3 Surface Area = 6 x S 2 S Doctor Wat s Mat Magic Book One

4 Voume and Surface Area of a Cyinder Cacuating te voume and surface area of a cyinder is easy if you use te information tat e earned about circes. r Te voume of te cyinder is equa to te area of te circuar end times te eigt. You remember tat to find te area of a circe e use te fooing reationsip (equation.) So te voume of a cyinder is cacuated using tis reationsip (equation.) Te surface area of a cyinder is aso easy to cacuate. Tere are to circuar end surfaces tat are cacuated using te same reationsip (equation) tat e just orked it: π d Te area of te vertica, curved surface is easy to cacuate if you visuaize unrapping it as toug it ere te paper abe on a grocery can. Since te unrapped edge is te circumference of te circe, and te vertica edge is te eigt of te cyinder, te area can be represented by te fooing reation (equation.) Doctor Wat s Mat Magic Book One

5 Tota Surface Area Te tota surface area is cacuated by combining te to circuar areas and te one unrapped area using te fooing reationsip (equation.) Let s sove a simpe probem using te voume of a cyinder reationsip. Let s suppose you ave a arge cyindrica container tat is partiay fu of paint. You oud ike to pour te paint into tree smaer containers tat are a te same size but you are not sure if tey can od a of te paint. It oud be nice to cacuate an anser before messing it te paint. You measure te sma containers and find tat eac one is 4 inces in diameter and 6 inces ig. 7 dia Te arge container is 7 inces in diameter and is fied to a dept of 5 inces it te paint. 4 dia 6 5 Our first task is to determine o muc paint is in te arge container. We kno tat te voume can be cacuated using te fooing reation. (equation.) Pugging in te numbers and using 3.14 for te vaue of π e get: V= 3.14 x 3.5 x 3.5 x 5 V = cubic inces So tis is te amount of paint tat e ave to pour into te tree smaer containers. Doctor Wat s Mat Magic Book One

6 Next, et s find te voume of one of te sma containers. We use te same reationsip (equation.) Pugging in te numbers e get: V = 3.14 x 2 x 2 x 6 V = We ave tree sma containers, so te tota voume tey can ande is: 3 x = Since tis is greater tan te voume of paint in te big container, e can go aead and pour te paint! You i run into many probems tat require you to cacuate voumes of objects. Remember te fooing pointers en you ande tese probems: First: Write don te dimensions of eac object in te probem. Sometimes te probem as a tist. In te exampe above, I coud ave been sneaky and aso given you te eigt of te arge container. For tat probem, te actua eigt is not important, since te can is partiay fied! Second: Carefuy cacuate te voume of eac object in te probem. Tird: Write don a reationsip tat describes te probem. For instance, in te probem above, e kne tat te tota of te tree sma voumes ad to be greater tan te voume of paint in te arge can. Doctor Wat s Mat Magic Book One