Lesson Dimensional Solids. Objectives. Classify 3-Dimensional solids Determine the Volume of 3-Dimensional solids. Student Name: Date:

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1 Student Name: Date: Cntact t Persn Name: Phne Number: Lessn -Dimensinal Slids Objectives Classify -Dimensinal slids Determine the Vlume f -Dimensinal slids

2 Authrs: Jasn March,.A. Tim Wilsn,.A. Editr: Graphics: Linda Shanks Tim Wilsn Jasn March Eva McKendry Natinal PASS Center OCES Genese Migrant Center 7 Lackawanna Avenue Munt Mrris, NY (585) (585) (fax) Develped by the Natinal PASS Center under the leadership f the Natinal PASS Crdinating Cmmittee with funding frm the Regin 0 Educatin Service Center, San Antni, Texas, as part f the Mathematics Achievement Success (MAS) Migrant Educatin Prgram Cnsrtium Incentive prject. In additin, prgram supprt frm the Opprtunities fr Success fr Out-f-Schl Yuth (OSY) Migrant Educatin Prgram Cnsrtium Incentive prject under the leadership f the Kansas Migrant Educatin Prgram.

3 In the last several lessns, we have discussed different types f gemetric figures. Our discussin has been limited t tw-dimensinal (-D) bjects. Hwever, we live in a three-dimensinal (-D) wrld. Nw let s discuss bjects that exist in the three-dimensinal wrld. Fr example, a rectangle is a -D bject. It nly has tw dimensins a base and a height. h b ut, what if we added a third dimensin t the rectangle? b (base) can als be represented by l (length). In a -D figure, h (height) can be called w (width). In a -D figure, w (width) might be called d (depth). h w b Here we have an bject made f three dimensins: base (b), ( height t (h),( and width (w). ( This shape is knwn as a rectangular prism. A rectangular prism is a three-dimensinal slid. It has tw parallel bases that are cngruent rectangles. In the figure drawn, the bases are shaded gray. This may be difficult t see n paper s let s think f sme real life examples f this slid. A shebx A textbk A refrigeratr -D bjects are ften called slids. Math On the Mve Lessn 1

4 The whle wrld is made f three-dimensinal slids. There are mre than just rectangular prisms. In fact, a prism can define many types f slids. A prism is a slid figure. It has tw parallel bases that are cngruent plygns. A prism can have bases shaped like any plygn. Example Classify the fllwing prisms. a) b) c) Slutin We knw these are all prisms, s we need t lcate the base and determine which plygn it is. a) The bases n this prism are shaded in gray. Each bases has five vertices and sides, s they are pentagns.. This means that the slid is a pentagnal prism. b) The bases f this prism have three vertices and sides. The plygn with three sides is a triangle, s this is a triangular prism. c) The bases n this prism have six vertices and sides. The six sided plygn is a hexagn, s this is a hexagnal prism. Ntice, when the base is a plygn with five r mre sides, the prism name ends in -al. We add -al t the name f the plygn. Pentagnal, Hexagnal, Heptagnal, Octagnal, Nnagnal, Decagnal Math On the Mve

5 Anther shape that has tw parallel, cngruent bases is a cylinder.. Hwever, a cylinder is nt cnsidered a prism. A cylinder is a slid with tw parallel, cngruent bases that are circles. The fllwing is an example f a cylinder Think ack Again, this may be hard t see n paper, s let s think f sme bjects that are shaped like a cylinder. A can f beans A marker A pipe A cylinder is nt a prism because its base is a circle, and a circle is nt a plygn. There are three dimensinal shapes that have nly ne base. Think f the ancient Egyptians and the tmbs they built fr their Pharahs.. The tmb shape is mre cmmnly knwn as a pyramid. A pyramid is a -D slid. Its base is a plygn, p and its sides are triangles that cnverge t a cmmn pint. The pyramids built by the ancient Egyptians are square pyramids.. Their bases are squares. Square Pyramid Triangular Pyramid Math On the Mve Lessn

6 Mst pyramids have a square r triangular base. Other kinds f bases are pssible, as well. A shape similar t a pyramid is a cne. A cne is a slid with ne circular base and ne vertex. The fllwing is an example f a cne. A cne is a familiar shape that we see all the time. Sme examples f cnes are: A traffic cne A megaphne An ice i cream cne The last -D shape we need t discuss has n base. This shape is knwn as a sphere. A sphere is a -Dimensinal slid that is perfectly rund. Every pint n a sphere is the same distance frm the center f the sphere. The fllwing is an example f a sphere. Math On the Mve 4

7 The sphere is the hardest shape t see n paper. Sme real life examples f a sphere include: A sccer ball A glbe (f the Earth) A baseball An range Nw we have discussed all the basic slids. Try classifying ing sme n yur wn! 1. Classify the fllwing slid figures. a) b) c) d) Math On the Mve Lessn 5

8 Nw that we knw abut -D slids, we can slve prblems using them. Yur friend, Daniela, just finished building a rectangular pl. She wants t knw hw much water she needs t fill it. She knws, but she des nt 1 kl 1 m knw hw many cubic meters are in her pl. 1.5 m 10 m 7 m A cubic meter is a measure f vlume. A killiter is a measure f capacity. Vlume is the amunt f space a slid takes up. Capacity is the amunt that an bject can hld. S if 1 kl 1 m, then 1 kl takes up 1 m f space. This is why Daniela needed kl f water, nt m f water, t fill her pl. f water, t fill her pl. Daniela shws yu a picture f her pl. Yu can help her find ut hw much water she needs. Her pl is a rectangular prism. T find ut hw much water she needs, yu need t find the vlume f the rectangular prism. Vlume is the amunt f space that a slid takes up. Vlume is measured in cubic units. Vlume is very similar t area. When yu wanted t find the area f rectangles, yu added up the square units. This rectangle has an area f 18 square units. Initially, we fund this by cunting up all the squares. If we wanted t find the vlume f a rectangular prism, we just add up all the cubes. A cube is a rectangular prism with 6 cngruent square faces. A standard six-sided sided die is an example f a cube Math On the Mve 6

9 The vlume f this rectangular prism is 18 cubic units,, because there are 18 cubes in ttal. Example Find the vlume f the fllwing f rectangular prisms a) b) c) Slutin T find the vlume f the given slids, we simply add up all the cubes. a) We can see all the cubes in this rectangular prism. The dimensins f the prism are 1 by 1 by 4. There are 4 ttal cubes in the prism, s the vlume is 4 cubic units,, r 4 units. b) In this prism, we cannt see all the cubes. We have t figure ut hw many cubes there are using reasning. The frnt face f the prism has 8 cubes in it. We can see that there are rws with 8 cubes in it. We can assume that there are 8 4 cubes. This means the vlume is 4 cubic units,, r 4 units. rws c) In this prism, we cannt see all the cubes. We can see that the frnt f face f the prism has 4 cubes in it. The prism als has tw rws with 4 cubes in it. We can assume there are 4 8 cubes. This means the vlume is 8 cubic units,, r 8 units. rws Remember that with rectangles, we can multiply the dimensins t find the area. What d yu ntice abut the dimensins f the rectangular prism and the vlume? Math On the Mve Lessn 7

10 In example a, the dimensins are 1 by 1 by 4. Think ack The vlume is cubic units Remember that with rectangles, dimensins In example b, the dimensins are by 4 by. were said base by height. With rectangular prisms, The vlume is 4 4 cubic units. 4 they are said length by width by height. In example c, the dimensins are by by. The vlume is 8 cubic units. The examples shw us that vlume is fund by multiplying the three dimensins f the prism. The vlume frmula fr a rectangular prism is V l w h Vlume length width height The three dimensins can be called a variety f names. We use length, width, and height as a standard fr the frmula. Math On the Mve 8

11 Let s lk back at the prblem with Daniela s pl. She gave us the dimensins f the pl. S, t find the vlume f the pl, we simply multiply all three dimensins. V m 1.5 m 10 m 7 m If 1 kl 1 m, Daniela needs needs 105 kl f water t fill the pl. Example Find the vlume f the fllwing cube. Slutin in. T find the vlume f the cube, we need t knw its three dimensins. Since a cube is made f cngruent squares, the lengths f all the edges are the same. S, the three dimensins f this cube are by by. The vlume is 7 in. Ntice that we multiplied the edge by itself three times. Remember that repeated multiplicatin is the same as using an expnent. S, the vlume frmula fr a cube is thelengthf needge fr a cube is ( ). The term cubed cmes frm the vlume f a cube. Math On the Mve Lessn 9

12 . Find the vlume f the fllwing rectangular prisms. a) b) km 1 km km c) d) 11 cm 1 cm 5 ft. 7 cm ft. 5 ft. Math On the Mve 10

13 The vlume f a rectangular prism is fund by multiplying all three dimensins. Hwever, there is anther frmula that we use fr all prisms. V h Where is the area f the base f the prism and h is the height. When we were finding the vlume f rectangular A Think ack bh is nt the same as prisms, we simply brke dwn the area f the base int the V h. We use the length and width. lwercase b when we are describing the length f the base. We use the uppercase when we are describing the area f the base. h l w The area f the base f this rectangular prism p is l w. This means the vlume f the rectangular prism is. V h l w h Nw that we knw this, we can use the new frmula t find the vlume f triangular prisms. Example Find the vlume f the fllwing triangular ar prism. m 4 m m Math On the Mve Lessn 11

14 Slutin First, we must find the base f the prism. Since it is a triangular prism, the base is the right triangle n the frnt side f the prism. m 4 m Next, we need t find the area f that triangle in frnt. 1 bh m ( )( ) Nw that we have the area f the base, we can substitute, r plug that int the frmula t find the vlume f the prism. V V h 6( ) 1 e careful when yu are plugging in values fr the frmula. We used as the height f the m m prism, because we already used and 4 t find the area f the base,. Think ack Area frmula fr a 1 triangle is A bh. Even thugh the cylinder is nt a prism, we can use the same frmula fr finding the vlume. This wrks because,, just like a prism, a cylinder has tw cngruent bases. Example Find the vlume f the t fllwing cylinder. (Rund yur answer t the nearest hundredth) m 9 m Math On the Mve 1

15 Slutin T find the vlume f the cylinder, we must use the vlume frmula. V A cylinder has tw circular bases, s t find the area f the base,,, we need ed t find the area f the circle. The circle has a radius f meters. We plug that value int the area frmula fr a circle. π r π 9π We will use.14 t estimate pi. 9π ( ) ( ) m Nw that we have fund the area f the base,,, we can plug that int the vlume frmula. V V h h ( 8.6)( 9) 54.4 m S, the vlume f the cylinder is apprximately 54.4 m. Remember, because.14 is an apprximate value f pi, any answer invlving it is apprximate. Think ack Area frmula fr a circle is A π r. When we are finding the length f an bject, the answer is in units (ft., in., m, km, etc.). When we are finding the area f an bject, the answer is in square units ( ft., in., m, km ). When we are finding the vlume f an bject, the answer is in cubic units ( ft., in., m, km ). Math On the Mve Lessn 1

16 Lastly, we need t determine the vlume f the slids with ne base. When we fund the area f -D triangles, we cmpared them t parallelgrams and fund their area t be 1 f a parallelgram. We will use a similar methd fr finding f the vlume f -D pyramids and cnes. A square pyramid is similar t a cube because they bth have a square base. Hwever, the pyramid nly has ne base, while the cube has tw. Square Pyramid Cube Cne Cylinder ecause we are wrking with -D slids,, the vlume f the pyramid is 1 the vlume f the cube. The vlume f the cne is 1 the vlume f the cylinder. The pyramid and the cube, as well as the cne and the cylinder, share the fllwing dimensins: the base,,, and the height, h. In a pyramid and a cne, the height is the perpendicular line frm the base t the vertex where all the edges meet, as shwn abve. S, the vlume frmula fr a pyramid (and a cne) is V 1 h Where represents the area a f the base, and h represents the height. Math On the Mve 14

17 Example Find the vlume f the fllwing cne. (Rund yur y answer t the nearest tenth) 8 in. 7 in. Slutin T find the vlume f the cne, we must use the frmula V 1 h. Remember that represents the area f the circular base. First, let s find.. We are given the diameter f the circular base, but the area frmula fr the circle requires the radius. The radius is half f the diameter, s the radius f the circle is the base,. π r π (.5) 1.5π We will use.14 t estimate pi. 1.5π ( ) in Nw we can use that fr the area f Calculatr Tip T square difficult numbers, we use a calculatr. Enter the number n the calculatr and hit the square buttn. x Nw that we knw the area f the base, we can plug that int the vlume frmula. Math On the Mve Lessn 15

18 V 1 h ( )( ) in. The prblem asked us t rund t the nearest tenth, s the vlume f the cne is V 10.6 in. Think ack T rund a number, lk at the number t the right f the place value we are asked t rund t. Then cmpare that number t 5. Try t slve the fllwing area prblems n yur wn.. Find the vlume f the fllwing slids. (Rund t the nearest whle number) a) Cne b) Cube 4 cm 4.5 cm km Math On the Mve 16

19 c) Square Pyramid d) Cylinder 8 15 m m Review 1. Highlight the fllwing definitins: a. rectangular prism b. prism c. cylinder d. pyramid e. cne f. sphere g. vlume h. cube. Highlight all the vlume frmulas in the lessn.. Highlight ight all the Fact bxes. 4. Highlight all the Think back bxes. Math On the Mve Lessn 17

20 5. Write ne questin yu wuld like t ask yur mentr, r ne new thing yu learned in this lessn. Practice Prblems Math On the Mve Lessn Directins: Write yur answers in yur math jurnal. Label this exercise Math On the Mve Lessn,, Set A and Set. Set A 1. Which tw types f slids have tw bases?. What d yu knw abut the dimensins f a cube?. Find the vlume f the fllwing slids. a) b) 11 ft. 5.5 ft. 8 ft. 8 m 5 m m Math On the Mve 18

21 c) d) 10 mm 0 mm cm 1 cm 18 cm Set 1. Draw the slid described. a) A cylinder b) A pyramid with a square base c) A cne. What is the nly slid that has n base?. If a cube has a vlume f 64, what is the length f ne edge? 1. a) Cne b) Rectangular prism (Cube) c) Triangular pyramid d) Octagnal prism. a) 150 units b) 10 km c) 77 cm d) 50 ft.. a) a) π ( 4 ) π ( 16) (.14)( 16) V V 1 ( 50.4 )( 4.5 ) 75.6 V 75 cm 50.4 cm b) V V 8 km Math On the Mve Lessn 19

22 c) units V V 1 ( 5 )( 8 ) 66.6 V 67 units d) π ( 15 ) π ( 5) (.14)( 5) m V ( 706.5)( 4) V 86 m End f Lessn Math On the Mve 0