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22. Architecture The Capitol Rotunda connects theHouse and the Senate sides of the U.S. Capitol.Complete the table. Round your answers to thenearest foot.Capitol Rotunda DimensionsHeightCircumference180 ft301.5 ftRadiusDiameterFOCUS ON HIGHER ORDER THINKING23. Multistep A museum groundskeeper is creating a semicircular statuarygarden with a diameter of 30 feet. There will be a fence around thegarden. The fencing costs 9.25 per linear foot. About how much will thefencing cost altogether?Work Area24. Critical Thinking Sam is placing rope lights around the edge of a circularpatio with a diameter of 18 feet. The lights come in lengths of 54 inches.How many strands of lights does he need to surround the patio edge?25. Represent Real-World Problems A circular path 2 feet wide has an innerdiameter of 150 feet. How much farther is it around the outer edge of thepath than around the inner edge?27. Persevere in Problem Solving Consider two circular swimming pools.Pool A has a radius of 12 feet, and Pool B has a diameter of 7.5 meters.Which pool has a greater circumference? How much greater? Justify youranswers.270Unit 4 Houghton Mifflin Harcourt Publishing Company26. Critique Reasoning A gear on a bicycle has the shape of a circle.One gear has a diameter of 4 inches, and a smaller one has a diameter of2 inches. Justin says that the circumference of the larger gear is 2 inchesmore than the circumference of the smaller gear. Do you agree? Explainyour answer.

LESSON9.2 Area of Circles?7.4.9.2Students will find the area ofa circle.ESSENTIAL QUESTIONHow do you find the area of a circle?EXPLORE ACTIVITY 1Exploring Area of CirclesYou can use what you know about circles and π to helpfind the formula for the area of a circle.STEP 1Use a compass to draw a circle and cut it out.STEP 2Fold the circle three times as shown to getequal wedges.STEP 3Unfold and shade one-half of the circle.STEP 4Cut out the wedges, and fit the piecestogether to form a figure that looks likea parallelogram.The base and height of the parallelogram relate to the parts of the circle.Radiusbase b the circumference of the circle, or Houghton Mifflin Harcourt Publishing CompanyHalf the circumferenceheight h theof the circle, orTo find the area of a parallelogram, the equation is A To find the area of the circle, substitute for b and hin the area formula.A bhA Reflect1. How can you make the wedges look more likea parallelogram?hSubstitute for b.A πrSubstitute for h.A πWrite using an exponent.Lesson 9.2271

Finding the Area of a CircleArea of a CircleMath On the SpotThe area of a circle is equal to π timesthe radius squared.my.hrw.comrA πr 2Remember that area is given in square units.EXAMPLE 1Math TalkMathematical ProcessesIf the radius increases by1 centimeter, how does thearea of the top of thebiscuit change?A πr2Use the formula.A π(4)2Substitute. Use 4 for r.A 3.14 42Substitute. Use 3.14 for π.A 3.14 16Evaluate the power.A 50.24Multiply.The area of the biscuit is about 50.24 cm2.Reflect2. Compare finding the area of a circle when given the radius with findingthe area when given the diameter.3.PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com272Unit 4Why do you evaluate the power in the equation before multiplying by π?YOUR TURN4.A circular pool has a radius of 10 feet. What is the areaof the surface of the water in the pool? Use 3.14 for π. Houghton Mifflin Harcourt Publishing Company Image Credits: ZigzagMountain Art/ShutterstockA biscuit recipe calls for the dough to be rolled out and circles to be cutfrom the dough. The biscuit cutter has a radius of 4 cm. Find the area ofthe top of the biscuit once it is cut. Use 3.14 for π.

EXPLORE ACTIVITY 2Finding the Relationship betweenCircumference and ArearYou can use what you know about circumference and area of circles tofind a relationship between them.Find the relationship between the circumference and area of a circle.r Start with a circle that has radius r.Solve the equation C 2πr for r.( )( )2A πSubstitute your expression for r in theformula for area of a circle.Remember: Becausethe exponent is outsidethe parentheses,you must apply it tothe numerator andto each factor of thedenominator.2A π22·Square the term in the parentheses.2·A 2· Houghton Mifflin Harcourt Publishing CompanyEvaluate the power.2Simplify.A ·Solve for C2.C2 4The circumference of the circle squared is equal to.Lesson 9.2273

EXPLORE ACTIVITY 2 (cont’d)Reflect5.Does this formula work for a circle with a radius of 3 inches? Show your work.Guided PracticeFind the area of each circle. Round to the nearest tenth if necessary. Use 3.14 for π.(Explore Activity 1)1.2.14 m3.20 yd12 mm4. A clock face has a radius of 8 inches. What is the area of the clockface? Round your answer to the nearest hundredth.5. A DVD has a diameter of 12 centimeters. What is the area of theDVD? Round your answer to the nearest hundredth.6. A company makes steel lids that have a diameter of 13 inches. Whatis the area of each lid? Round your answer to the nearest hundredth.Find the area of each circle. Give your answers in terms of π.(Explore Activity 2)7. C 4π8. C 12πA A π9. C 2A 10. A circular pen has an area of 64π square yards. What is thecircumference of the pen? Give your answer in terms of π.(Explore Activity 2)?ESSENTIAL QUESTION CHECK-IN11. What is the formula for the area A of a circle in terms of the radius r?274Unit 4 Houghton Mifflin Harcourt Publishing CompanySolve. Use 3.14 for π. (Example 1)

NameClassDate9.2 Independent PracticePersonalMath Trainermy.hrw.com12. The most popular pizza at Pavone’s Pizzais the 10-inch personal pizza with onetopping. What is the area of a pizza with adiameter of 10 inches? Round your answerto the nearest hundredth.13. A hubcap has a radius of 16 centimeters.What is the area of the hubcap? Roundyour answer to the nearest hundredth.16 cmOnlineAssessment andIntervention17. Multistep The sides of a square field are12 meters. A sprinkler in the center of thefield sprays a circular area with a diameterthat corresponds to a side of the field.How much of the field is not reached bythe sprinkler? Round your answer to thenearest hundredth.18. Justify Reasoning A small silver dollarpancake served at a restaurant has acircumference of 2π inches. A regularpancake has a circumference of 4π inches.Is the area of the regular pancake twice thearea of the silver dollar pancake? Explain. Houghton Mifflin Harcourt Publishing Company14. A stained glass window is shaped like asemicircle. The bottom edge of the windowis 36 inches long. What is the area of thestained glass window? Round your answerto the nearest hundredth.15. Analyze Relationships The point (3, 0)lies on a circle with the center at the origin.What is the area of the circle to the nearesthundredth?16. Multistep A radio station broadcasts asignal over an area with a radius of 50miles. The station can relay the signal andbroadcast over an area with a radius of75 miles. How much greater is the area ofthe broadcast region when the signal isrelayed? Round your answer to the nearestsquare mile.19. Analyze Relationships A bakery offersa small circular cake with a diameter of8 inches. It also offers a large circular cakewith a diameter of 24 inches. Does the topof the large cake have three times the areaof that of the small cake? If not, how muchgreater is its area? Explain.Lesson 9.2275

2C20. Communicate Mathematical Ideas You can use the formula A 4π tofind the area of a circle given the circumference. Describe another way tofind the area of a circle when given the circumference.21. Draw Conclusions Mark wants to order a pizza. Whichis the better deal? Explain.Donnie’s Pizza PalaceDiameter (in.)1218Cost ( )102022. Multistep A bear was seen near a campground. Searchers weredispatched to the region to find the bear.a. Assume the bear can walk in any direction at a rate of 2 miles perhour. Suppose the bear was last seen 4 hours ago. How large an areamust the searchers cover? Use 3.14 for π. Round your answer to thenearest square mile.b. What If? How much additional area would the searchers have tocover if the bear were last seen 5 hours ago?23. Analyze Relationships Two circles have the same radius. Is thecombined area of the two circles the same as the area of a circle withtwice the radius? Explain.24. Look for a Pattern How does the area of a circle change if the radius ismultiplied by a factor of n, where n is a whole number?25. Represent Real World Problems The bull’s-eye on a target has adiameter of 3 inches. The whole target has a diameter of 15 inches. Whatpart of the whole target is the bull’s-eye? Explain.276Unit 4Work Area Houghton Mifflin Harcourt Publishing CompanyFOCUS ON HIGHER ORDER THINKING

LESSON9.3?Area of CompositeFigures7.4.9.3Students will find the area ofcomposite figures.ESSENTIAL QUESTIONHow do you find the area of composite figures?EXPLORE ACTIVITYExploring Areas of Composite FiguresAaron was plotting the shape of his garden on grid paper. While it was anirregular shape, it was perfect for his yard. Each square on the gridrepresents 1 square meter.A Describe one way you can find the area of this garden.B The area of the garden issquare meters. Houghton Mifflin Harcourt Publishing CompanyC Compare your results with other students. What other methodswere used to find the area?D How does the area you found compare with the area found usingdifferent methods?Reflect1. Use dotted lines to show two differentways Aaron’s garden could be dividedup into simple geometric figures.Lesson 9.3277

Finding the Area of a Composite FigureMath On the Spotmy.hrw.comA composite figure is made up of simple geometric shapes. To find the area ofa composite figure or other irregular-shaped figure, divide it into simple,nonoverlapping figures. Find the area of each simpler figure, and then addthe areas together to find the total area of the composite figure.Use the chart below to review some common area formulas.ShapeArea FormulatriangleA 12 bhsquareA s2rectangleA ℓwparallelogramA bhtrapezoidA 12 h(b1 b2)EXAMPLE 1Find the area of the figure.STEP 1my.hrw.comSTEP 210 cm1.5 cm3 cm4 cm1.5 cm7 cmbase1 7 cmheight 1.5 cmheight 1.5 cmUse the formula.Use the formula.2 cmbase2 10 cmA 10 · 1.5The top base of thetrapezoid is 10 cm1since it is the sameA 2 h(b1 b2)length as the base ofA 12 (1.5)(7 10) the parallelogram.A 15A 12 (1.5)(17) 12.75The area of theparallelogram is 15 cm2.The area of the trapezoid is 12.75 cm2.Add the areas to find the total area.A 15 12.75 27.75 cm2The area of the figure is 27.75 cm2.Unit 4Area of the Trapezoidbase 10 cmA bh2782 cm1.5 cm7 cm4 cmFind the area of each shape.Area of the ParallelogramSTEP 310 cm1.5 cmSeparate the figure into smaller,familiar figures: a parallelogram 3 cmand a trapezoid. Houghton Mifflin Harcourt Publishing CompanyAnimatedMath

YOUR TURNFind the area of each figure. Use 3.14 for π.2.PersonalMath Trainer3.2 ft3 ft3 ft3 ftOnline Assessmentand Intervention8 ftmy.hrw.com4 ft10 m8 ft10 mUsing Area to Solve ProblemsEXAMPL 2EXAMPLEMath On the SpotA banquet room is being carpeted. A floor plan of theroom is shown at right. Each unit represents1 yard. The carpet costs 23.50 per square yard.How much will it cost to carpet the room?STEP 1Separate the composite figure into simplershapes as shown by the dashed lines: aparallelogram, a rectangle, and a triangle.STEP 2Find the area of the simpler figures. Count units to find thedimensions.Parallelogram Houghton Mifflin Harcourt Publishing Companymy.hrw.comSTEP 3RectangleTriangleA bhA ℓwA 12 bhA 4·2A 6·4A 12 (1)(2)A 8 yd2A 24 yd2A 1 yd2Math TalkMathematical ProcessesDescribe how you canestimate the cost tocarpet the room.Find the area of the composite figure.A 8 24 1 33 square yardsSTEP 4Calculate the cost to carpet the room.Area · Cost per yard Total cost33 · 23.50 775.50The cost to carpet the banquet room is 775.50.Lesson 9.3279

YOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com4. A window is being replaced with tinted glass.The plan at the right shows the design of thewindow. Each unit length represents 1 foot.The glass costs 28 per square foot. How muchwill it cost to replace the glass? Use 3.14 for π.Guided Practice1. A tile installer plots an irregular shape on grid paper. Each squareon the grid represents 1 square centimeter. What is the area of theirregular shape? (Explore Activity, Example 2)STEP 1Separate the figure into a triangle, aand a parallelogram.STEP 2Find the area of each figure.triangle:STEP 3cm2; rectangle:,cm2; parallelogram: Find the area of the composite figure:The area of the irregular shape is cm2 cm2cm2.12 cm2. Show two different ways to divide the composite figure.Find the area both ways. Show your work below. (Example 1)9 cm8 cm18 cm20 cm3. Sal is tiling his entryway. The floor plan is drawn on a unit grid.Each unit length represents 1 foot. Tile costs 2.25 per square foot.How much will Sal pay to tile his entryway? (Example 2)?ESSENTIAL QUESTION CHECK-IN4. What is the first step in finding the area of a composite figure?280Unit 4 Houghton Mifflin Harcourt Publishing Company9 cm

NameClassDate9.3 Independent PracticePersonalMath Trainermy.hrw.com5. A banner is made of a square and asemicircle. The square has side lengths of26 inches. One side of the square is alsothe diameter of the semicircle. What is thetotal area of the banner? Use 3.14 for π.8. A field is shaped like the figure shown.What is the area of the field? Use 3.14 for π.8m8m6. Multistep Erin wants to carpet the floorof her closet. A floor plan of the closet isshown.10 ft4 ft3 ft6 fta. How much carpet does Erin need? Houghton Mifflin Harcourt Publishing Companyb. The carpet Erin has chosen costs 2.50per square foot. How much will it costher to carpet the floor?7. Multiple Representations HexagonABCDEF has vertices A(-2, 4), B(0, 4), C(2, 1),D(5, 1), E(5, -2), and F(-2, -2). Sketch thefigure on a coordinate plane. What is thearea of the hexagon?OnlineAssessment andIntervention8m9. A bookmark is shaped like a rectangle witha semicircle attached at both ends. Therectangle is 12 cm long and 4 cm wide. Thediameter of each semicircle is the widthof the rectangle. What is the area of thebookmark? Use 3.14 for π.10. Multistep Alex is making 12 pennants forthe school fair. The pattern he is using tomake the pennants is shown in the figure.The fabric for the pennants costs 1.25 persquare foot. How much will it cost Alex tomake 12 pennants?1 ft3 ft1 fty2-4 -2O2411. Reasoning A composite figure is formedby combining a square and a triangle. Itstotal area is 32.5 ft2. The area of the triangleis 7.5 ft2. What is the length of each side ofthe square? Explain.-4Lesson 9.3281

Work AreaFOCUS ON HIGHER ORDER THINKING12. Represent Real-World Problems Christinaplotted the shape of her garden on graph paper.She estimates that she will get about 15 carrotsfrom each square unit. She plans to use the entiregarden for carrots. About how many carrots can sheexpect to grow? Explain.13. Analyze Relationships The figure shownis made up of a triangle and a square. Theperimeter of the figure is 56 inches. What isthe area of the figure? Explain.10 in.28 in.15. Persevere in Problem Solving The design for thepalladium window shown includes a semicircularshape at the top. The bottom is formed by squaresof equal size. A shade for the window will extend4 inches beyond the perimeter of the window,shown by the dashed line around the window.Each square in the window has an area of 100 in2.a. What is the area of the window? Use 3.14 for π.b. What is the area of the shade? Round your answer to the nearestwhole number.282Unit 415 in. Houghton Mifflin Harcourt Publishing Company14. Critical Thinking The pattern for a scarf isshown at right. What is the area of the scarf?Use 3.14 for π.10 in. 8 in.

LESSON9.4?Solving Surface AreaProblemsESSENTIAL QUESTION7.4.9.4Students will find the surfacearea of a figure made up ofcubes and prisms.How can you find the surface area of a figure made up of cubesand prisms?EXPLORE ACTIVITYModeling Surface Area of a PrismThe surface area of a three-dimensional figure is the sum of the areas ofall its surfaces. You know how to use the net of a figure to find its surfacearea. Now you will discover a formula that you can use.8 mmbase 15 mm20 mm20 mm side8 mmfrontsideback8 mm 15 mm15 mmbase 15 mmA The lateral area L of a prism is the area of all faces except the bases. Houghton Mifflin Harcourt Publishing CompanyL 2() 2(B The area B of each base is) .C The surface area S of the prism is the sum of the lateral area L and thetotal area of the bases, or.Reflect1. Analyze Relationships Use the net above to answer this question:How does the product of the perimeter P of the base of the prism andthe height h of the prism compare to the lateral area L?2. Critical Thinking How can you express the surface area S of theprism in terms of P, h, and B? Use your answer to Question 1.Lesson 9.4283

Finding the Surface Area of a PrismGiven a prism’s dimensions, you can use a formula to find the surface area.Surface Area of a PrismMath On the SpotThe surface area S of a prism with base perimeter P, height h, andbase area B is S Ph 2B.my.hrw.comEXAMPLE 1My NotesErin is making a jewelry box of wood in theshape of a rectangular prism. The jewelry boxwill have the dimensions shown. She plans tospray paint the exterior of the box. How manysquare inches will she have to paint?6 in.15 in.STEP 1Make a sketch of the box. Drawing adiagram helps you understand andsolve the problem.STEP 2Identify a base, and find its area and perimeter.12 in.Any pair of opposite faces can be the bases. For example, you canchoose the bottom and top of the box as the bases.B ℓ w 12 15 24 30 180 square inches 54 inchesIdentify the height, and find the surface area.The height h of the prism is 6 inches. Use the formula to find thesurface area.Math TalkMathematical ProcessesHow can you express theformula for the surface areaS of a rectangular prism interms of its dimensionsℓ, w, and h?S Ph 2BS 54(6) 2(180) 684 square inchesErin will have to spray paint 684 square inches of wood.YOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com284Unit 43. A brand of uncooked spaghetti comes in a box that is a rectangular prismwith a length of 9 inches, a width of 2 inches, and a height of 1 12 inches.What is the surface area of the box? Houghton Mifflin Harcourt Publishing CompanySTEP 3P 2(12) 2(15)

Finding the Surface Area of aComposite SolidA composite solid is made up of two or more solid figures. To find the surfacearea of a composite solid, find the surface area of each figure. Subtract any areanot on the surface.EXAMPL 2EXAMPLEMath On the Spotmy.hrw.comProblemSolvingDaniel built the birdhouse shown. What wasthe surface area of the birdhouse before thehole was drille

Math Trainer Online Assessment and Intervention Personal my.hrw.com Math On the Spot my.hrw.com 14Use the formula. ft 11 cm Finding Circumference The ratio of the circumference to the diameter is the same for all circles. This ratio is called π or pi, and you can approximate it as 3.14 o