Section 12.1 Explore Solids. ., that enclose a single region of space. An of a

Transcription

1 GEOMETRY Chpt. 12

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3 Section 12.1 Explore Solids A is a solid that is bounded by polygons, called., that enclose a single region of space. An of a polyhedron is a line segment formed by the intersection of two faces. A of a polyhedron is a point where three or more edges meet. Plural form: polyhedra or polyhedrons T e.ÿyg ÿ_.g ff So I i ds Polyhedla Nol Polyhedra (yltndcr ÿ'one Pyramid +phz+e A polyhedron is polygons. if all of its faces are congruent regular - all vertices point towards the outside. - at least one vertex points inside. regulal (orlvex rtonregulal (Oi%ÿTaVC 1

4 There are five regular polyhedra, ca{led Tetrahedron - Cube - Octahedron - Dodecahedron - Icosahedron - Classify Solids To name a prism or a pyramid, use the of the. The two ÿ_ of a prism are congruent polygons in para!{el planes. The ÿ of a pyramid is a polygon. 2

5 Theorem: Euler's Theorem The number of faces(f), vertices(v), and edges(e) of a polyhedron are related by the formula F + V = E + 2. Example: Faces: Vertices: Edges: Cross Sections: The intersection of the plane and the solid is called a The diagram to the right shows that an intersection of a plane and a triangular pyramid is a xaÿ: Describe the shape formed by the intersection of the plane and the cubes. 3

6 Name: Date: MDL Practice W Per; In # 1-3, identify the number and types of polygons that are faces of each polyhedron. In #4-7, given each polyhedron & its net, label the remaining vertices. 4) o v 5) ÿÿ MÿT N S A B F M N 4

7 6) 7) B C l In 8 & 9, draw the net for each solid and find the total surface area of each solid, 8) i\ 9) r = 3 inches 5

8 Section 12.2 Surface Area of Prisms & Cylinders A is a polyhedron with two congruent faces called The other faces are called The segments connecting the vertices are - the sum of the areas of all its faces. Also known as - the sum of the areas of its lateral faces. hasÿ laÿeÿaÿ A two-dimensional representation of the faces is called a This is an example of a net of a rectangular prism. Example: Draw the net of a pentagonal prism. 6

9 Formula for Right Prism LA = TA or SA = or where P = Perimeter of the base h = height of the prism B = Area of the base Example: Find the LA & SA(TA) of the rectangular prism. 3 in, *Choose one side to be your base and use that throughout! You can't use a different base for LA and SA(TA). 7 in, Example: Find the LA & SA(TA) of the triangular prism. 4 ft ft 7

10 A parallel planes. The distance between its bases. is a solid with congruent circular bases that lie in of the cylinder is the perpendicular Formula for Riqht Cylinder: bÿ LA = TA or SA = or where ÿ = r = radius of the base h = height of the cylinder An example of a net of a cylinder Example: Find the LA & SA(TA) of the right cylinder. 24 mm -60 mm 8

11 Namÿ MDL Date section 12.2 Period _ Find the lateral area & surface area of each prism or cylinder, Write formulal Don't forget the units! Y 2cm 8cm 6ftÿ.ÿ" ÿ/ 2Oft 8 ft LA = _.. LA = LA = 5A = ÿ 5A = SA'= 12 cm r=14 mm 4) /,,,,-/ 5)... ;-'7, #, ## ' #S# ### h:27 mm. 6) S 4 In #,., #ÿ.i = #S a h--6 8 m Pÿrimeter of trapÿzokl = 36 cm 5 in I f LA = LA = LA : SA = 5A = 5A = 9

12 8) Draw the net for the solid below, 9) Draw the net for the solid berow, In #10-12, find the lateral area and surface area of each. SHOW AlL WORK & DOÿT FORGET THE UNITS! 11) ix- 26 m 14 fn LA = LA = 5A = 5A = 12) Find the lateral area and surface area of a prism with an equilateral triangle base with side of 10 ÿ/3 cm and height 15 cm. LA = SA = 10

13 Section 12.3 Surface Area of Pyramids & Cones A is a polyhedron in which the base is a polygon & the lateral faces are triangles with a common vertex, called the of the The intersection of two lateral faces is a The intersection of the base and a lateral face is a A has a regular polygon for a base. The of a regular pyramid is the height of a lateral face of the regular pyramid. Regular Pyramid Formula for Reqular Pyramid: LA = TA or SA = or where P = Perimeter of the base = slant height B = Area of the base Example: Find the LA & SA(TA) of the regular pyramid. 11

14 A plane as the base. The base. has a circular base and a that is not in the same The of the base is the radius of the cone. is the perpendicular distance between the vertex and the hÿ Right Cone Formula for Riqht Cone: LA = TA or SA = or where ÿ = r = radius of the base = slant height Example: Find the LA & SA(TA) of the right cone. 12

15 Name: MDL Per: Date: WS Draw the net of each solid and label each part with the correct length, 4m 1) ÿ 2) I0 m s) 1 1 } l I S nl / m 7m 4) Determine whether each is a prism or a pyramid and give the name of each, b, ÿ c, 13

16 Find the lateral area and surface area of each right prism and right cylinder. Write the formula, show work and round answers to the nearest tenth. 5..,4. 6. ' J g Cm l'l- " '1/6 cm 12 cm 60 yd Formula Lateral Area Surface Area Formula Lateral Area Surface Area, v 17 in I11 Formula Lateral Area Surface Area Formula Lateral Area Surface Area 14

17 Find the lateral area and surface area of each pyramid and cone, Write the formula, show work and round a nswers to the nearest tenth, 9, ÿ in 8 in 4 ÿnl Formula Lateral Area Surface Area Formula Lateral Area Surface Area 11, 12, 8,2 yd 7 yd Formula Lateral Area Surface Area Formula Lateral Area Surface Area 15

18 Per: Date: 12, Review #1 1ÿ) Draw the ne, t ÿor thÿ square pyramid. 2) Drÿw the.ÿ1 for the pÿtagoÿal pyÿmid. 9 cm 40 cm LA = LA = 5A - SA = []33 cm cm 16 cm LA LA = SA "- SA = 10 16

19 Write the formula for each, show all work and don't forget your units /ÿ1? in, 5/ÿ/ÿ 8, 16 iÿ LA= SA= LA= SA= 9. A regular pyramid has a hexagonal base with a base edge of 8 meters and a height of 10 meters. What is the surface area of the pyramid? SA= 10. A right cone has a diameter of 16.9 yards and a height of yards. What is the surface area of the cone? SA= 17

20 Section 12.4 Volume of Prisms & Cylinders The of a solid is the number of contained in its interior. Remember: Volume is measured in cubic units ex: Formula of a Prism Formula for Volume of a Cylinder _ V _ where B = Area of the base h = height of the prism where r = radius of the base h = height of the cylinder... B ÿf2 Vÿ Bh Example: Find the volume of the solid, Right trapezoidal prism Right cylinder 18

21 Section 12.5 Volume of P'!ramids & Cones Formula for Volume of a Pyramid Formula for Volume of a Cone V.ÿ V where B = area of the base h = height of the pyramid where r = radius of the base h = height of the cone Example: Find the volume of each solid. Right Pyramid Right Cone 14m 19

22 Worksheet 12.4 Name Volume of Prisms and Cylinders Date Period Find the volume of each figure. Round your answers to the nearest tenth, if necessary. 1) 2) 4 ft 7km 3 ft 8 km 3.5 ft 3) 6 cm 4) 5 cÿ@ore 7 cm 6cm 5) 3 ft 3 ft 5 ÿ fl 20

23 7) 4m 8) 3mi 4mi 4.3 8ÿ 5 mi 3m 5mi 9) lo) 7 km 8 km lo ft 1 I 1 9 km 11) A cylinder with a radius of 4 yd and a height of 5 yd, 12) A square prism measuring 6 km along each edge of the base and 5 km tall. A hexagonal prism 5 yd tall with a regular base measuring 5 yd on each edge and an apothem of length 4,3 yd. 14) A trapezoidal prism of height 6 km. The parallel sides of the base have lengths 5 km and 3 km, The other sides of the base are each 2 km. The trapezoid's altitude measures 1.7 km. 21

24 Worksheet 12.5 Name Volume of Pyramids and Cones Date Period Find the volume of each figure. Round your answers to the nearest tenth, if necessary. 1) 7 mi 2 mi 2) 4mi 4mi 5mi ;) 12 cm 4) n 11 cm 11 cm 2 in 5 in 5) 6) 1 yd 12 yd 6m 8.3 yd 5.2m 22

25 7) 12 ft m i 8 ft 10 ft 9) yd 10) mi yd 6.2 yd 6 mi 7 mi l l) A square pyramid measuring 10 yd along each edge of the base with a height of 6 yd. 12) A pyramid 5 m tall with a right triangle for a base with side lengths 6 m, 8 m, and 10 m. 13) A cone with radius 4 m and a height of 12 m. 14) A hexagonal pyramid 11 ft tall with a regular base measuring 6 ft on each side and an apothem of length 5.2 ft. 23

26 Section 12.6 Surface Area & Volume of Spheres A is the set of all points in space equidistant from a given point. This point is called the of the sphere. A of a sphere is a segment from the center to a point on the sphere. A sphere is a segment whose endpoints are on the sphere. A of a of a sphere is a chord that contains the center. eÿheÿ aÿjius diamÿeÿ If a plane intersects a sphere, the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a of the sphere. The circumference of a great circle is the circumference of the sphere. Every great circle of a sphere separates the sphere into two congruent halves called Formula for a Sphere: TA or SA = V = where r = radius of the sphere 24

27 Example: Find the SA(TA) & Volume of a sphere with a radius of 8 cm. Example: Find the radius of a sphere that has a surface area of 200ÿ m2. Example: Find the volume of a sphere where its great circle has a circumference of 8ÿ in. Example: Find the volume of a composite solid. 25

28 Name: Per: Date: MDL WS 12.6 avÿ alli answers in EXACT forml iwrÿte foÿuÿaÿ ShOW work & don t foÿ 1) Write the formulas down for surface area and volume of a sphere. 5A = V-" 2) Find the surface area and volume of a spherÿ with diameter 100 cm. 5A = V= 3) Find the surface area and volume of a spherÿ if the great circle has a circumference of cm. SA = V= 4) Find the surface area and volume of a sphere if its radius is 12 inches. 5A = V= 5) Find the surface area and volume of a sphere if the great circle has an area of ft2, 5A= V= In #6ÿ11, describe each object as a model of a circle, sphere or neither. 6) tennis ball can 7) pancake 8) sun 9) basketball rim 10) globe 11) lipstick container 26

29 12) Find the surface area and volume of a sphere with a radius of 25 inches, SA = V= 3) ÿ 14) Total Volume Total Volume frÿ, Total Volume 27

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31 12. Section Congruent Explore and Similar Similar Solids Solids Two solids of the same type with equal ratios of corresponding linear measures, such as heights or radii, are called The common ratio is called the of one solid to the other solid. Any two cubes are similar, as well as any two spheres. SfmiJar ÿ,yitndetÿ Nonÿtmilar ÿ:yltnderÿ Example: Identify similar solids Tell whether the given rectangular prism is similar to the right rectangular prism shown at the right. A) B) 28

32 Similar Solids Theorem: If two similar solids have a scale factor of a:b, then corresponding areas have a ratio of az:bz, and corresponding volumes have a ratio of a3:b3. ri a V! aÿ Examÿale: Use the scale factor of similar solids Packaging: The cans shown are similar with a scale factor of 87:100. Find the surface area and volume of the larger can. S i! 29

33 Per: Date: WS 12.7 In #1-6, determine if each pair of solids is similar, coÿruentor neither, Show howyou deterÿined tnsl 1) z) 3) 48 m grn 2 ' 12 yd 16m 6 I lore 12 yd 4) 5) 3m 6) 11 cm 12.rn i W I"r 14 om 7 om om For #7-10, use the right rectangular prisms at the right. The two right rectangular prisms at the right are similar. 7) Find the ratio of the perimeters of the bases. 8) What is the ratio of their surface areas? 9) What is the ratio of their volumes? 10) Suppose the volume of the smaller prism is 60 in3, Find the volume of the larger prism, 30

34 11) The scale factor of a model car to an actual car is 1:16. Find the following: (SHOW WORld) a) If the model has a length of 12 Inches. What is the length of the actual car;) b) Each tire of the model has a circumference of 7.25 inches, What is the circumference of the tire on the actual car? c) The front windshield on the model as an AREA of 3,5 square inches. What is the corresponding area on the actua! car? d) The models' engine has a,1v.olume of 2 cubic inches. Find the volume of the actual engine, In #12 & 13, find the surface area and volume. 12) Ie ft Total Volume Total Volume 14) Using a scale factor of 1:2, find the surface area and volume of a solid similar to the solid in #12. 15) Using a scale factor of 2:5, find the surface area and volume of a solid similar to the solid in #13, 31

35 Name: Date: Period: MDL Geometry Ch. 12 Review WS SHOW ALL WORK!! Don't forget your units! Leave answers in terms of **You will also need to know how to find the total surface area and volume of a figure made up of combined solids. In # 1-4, decide whether each situation is characteristic of total surface area or volume.. You want to fill up a jar with lemonade. 2. You want to paint a railroad car.. You want to ice a birthday cake.. You want to fill up a swimming pool with water.. Name the following using the pyramid at the right. Vertex of the pyramid: A tateral edge: A lateral face: BaSe; N t4 L In # 6-8, draw the net of each solid. Then name the solid. 6, Two cones are similar. If the ratio of their radii is 3:4, then a) What is the ratio of their surface areas? b) What is the ratio of their volumes? 32

36 10. Find the total surface area and volume of a hemisphere whose great circle has an area of 100 ÿtf!2 Total V 11. Find the total surface area of a cylindrical water tank that is 20 meters tall and has a diameter of 14 meters. Total 12. How much water will it take to fill the water tank from question #117 ' Water 13. Calculate the slant height, height, lateral area, total surface area, and volume of the right pyramid below. 14. Calculate the slant height, lateral area, total surface area, and volume of the right cone below. Loterol Totol Volume 33

37 cm 12 in 25ÿm Lateral Total Volume Lateral Total Volume 17. If a spherical balloon has a great circle with a circumference of 16 ÿ inches, how much air is needed to fill it? Air 18. The volume of the larger prism is 128 cm3. If the prisms are similar, what is the volume of the smaller prism? 19. The surface area of the larger prism is 328 cm2. If the prisms are similar, what is the surface area of the smaller prism? 20. Find the volume of the hexagonal pyramid with base area of 25 in 2 and height of 18 in. 21. Find the volume of the hexagonal prism below. Remember to find the area of the hexagon first! 6 in,ÿ 34

38 How Dimensions Affect Perimeter, Area and Volume Perimeter Draw a rectangle with a width of 3 and a length of 5. Find the perimeter. P= Now draw a rectangle where you multiply each side of the previous rectangle by a factor of 2, Find the perimeter of the enlarged rectangle. P= Since the sides were increased by a factor of factor of, the perimeter has increased by a Area Draw a rectangle with a width of 4 and a length of 7. Find the area. A= Now draw a rectangle where you multiply each side of the previous rectangle by a factor of 3. 35

39 Name: MDL Per: Date: Chapter 12 Review 2 SHOW ALL WORK! Don't forget to fnclude units in your answer! Leave ÿ in the answer where appropriate! 1) Write the formulas for the following: area of a triangle area of a rectangle area of a circle area of a regular polygon area of a trapezoid circumference of a circle 2) Name the following using the polyhedron at the right. Name 2 edges: Name a lateral face: Name the bases: ÿ & Name the vertices: E D 3) Draw an example of a pyramid. 4)Draw an example of a prism. 5) Name two differences between pyramids and prisms, 6) Using the views of a solid figure given below, draw the back view of the figure. top viow loft view front viow right view back view 38

40 7) Usiÿ3 the views of a solid figure given below, draw the back view of the figure. top view leaÿ view front view right view back view 8) Draw the net of each solid pictured below. 9. Find the surface area and volume of the prism, SA= 19 cm V= 12 crn 10) The base of a prism is a regular hexagon that measures 5 cm on each side. The prism has a height of 12 cm. What is its lateral area? 39

41 11) The base of a prism is a regular hexagon that measures 8 cm on each side. The prism has a height of 26 cm. What is its lateral area? 12) Find the surface area of a cylindrical water tank that is 5 meters tall and has a diameter of 8 meters, 13) Find the surface area of a cylindrical water tank that is 15 meters tall and has a diameter of 12 meters. 14) Find the lateral area and surface area of the cylinder below. r=b m 11m Lateral Total Volume 15) Find the lateral area and surface area of the cylinder below. r=6 m Lateral Total Volume 16) Find the surface area of a square pyramid if the side length of the base is 7 cm and the slant height is 13 cm. 17) Find the surface area of a square pyramid if the side length of the base is 13 cm and the slant height is 8 cm. 40

42 18. Find the lateral area and surface area of the right cone below. Lÿ LA= SA= iÿ LA= SA= ***************Work problems 1-8 in practice workbook sec

43 STAARTM State of Texas Assessments of Academic Readiness A bh = 1 2 A A = 1 d d A = 1 ( b + b ) h ap = 1 2 S = 1 P 2 l S = 1 2 Pl + B l l V = 1 B 3 h V = r

44 STAARTM State of Texas Assessments of Academic Readiness Ax + By = C B A C 2x 60 x x 2 45 x 30 x 3 45 x