ACCELERATED MATHEMATICS CHAPTER 11 AREA AND PERIMETER TOPICS COVERED:

Transcription

1 ACCELERATED MATHEMATICS CHAPTER AREA AND PERIMETER TOPICS COVERED: Perimeter of polygons Area of rectangles and squares Area of parallelograms Area of triangles Area of trapezoids

2 Activity -: Accelerated Mathematics Formula Chart Perimeter Rectangle P = ( l + w) Circumference Circle C = π r or C = π d Area Rectangle A = bh Parallelogram A = bh Triangle bh A = or A = bh Trapezoid A = ( b + b ) h Circle A = π r Surface Area (8 th grade only) Lateral Total Prism S = Ph S = Ph + B Pyramid S = Pl Cylinder S π rh S = Pl + B S = π rh + π r = Volume Triangular prism V = Bh Rectangular prism V = Bh Cylinder V = π r h or V = Bh Pyramid or Cone V = Bh 3 (8 th grade only) Sphere 4 3 V = π r 3 (8 th grade only) Pi π 3.4 or π 7 Pythagorean Theorem a + b = c (8 th grade only) Customary Length mile = 760 yards yard = 3 feet foot = inches Customary Volume/Capacity pint = cups cup = 8 fluid ounces quart = pints gallon = 4 quarts Customary Mass/Weight ton =,000 pounds pound = 6 ounces Metric Length kilometer = 000 meters meter = 00 centimeters centimeter = 0 millimeters Metric Volume/Capacity liter = 000 milliliters Metric Mass/Weight kilogram = 000 grams gram = 000 milligrams Time year = months year = 5 weeks week = 7 days day = 4 hours hour = 60 minutes minute = 60 seconds

3 AREA OF TRIANGLES and QUADRILATERALS A = s OR A = l w Area of a square A = l w Area of a rectangle A = b h Area of a parallelogram Area of a trapezoid Area of trapezoid A = ( b + b ) h OR Area of triangle bh A = bh OR A = A = ( b + b ) h

4 Activity -: Perimeter Perimeter: The distance around the outside of a figure. Per means around. Meter means measure. Thus, the perimeter of a figure is the measure around it. Classify each shape by giving the most specific name possible. Then find the perimeter of each figure cm 0 cm 4 ft 4 ft 4 m 3.5 m 5 m 8 cm 9 ft 8.5 m.0 km m 6..9 km 0.9 m.4 km.6 km 0.9 m 0.9 m.8 m Regular polygon.5 km.8 m Find the perimeter of each rectangle..5 ft cm 5 ft 40 cm 8. m 4 in 5 m Find the missing part of each rectangle. 0. P = 60 mm L = 3 mm W =. P =.8 km W = 4.7 km L =. Find the perimeter of a sheet of typing paper 8.5 in. wide and in. long. 3. Which of the following CANNOT be used to find the perimeter of a square with side length s? A. s + s + s + s B. s + s C. 4s D. s s Peter wants to find the perimeter of the isosceles trapezoid shown below. Which equation could Peter use to find P, the perimeter of the trapezoid? 4. 8 inches 5 inches 4 inches A. P = B. P = ( 5) C. P = (8 + 4) 4 D. P =

5 Activity -3: Perimeter Two rectangles are shown below. The value of x is the same for both rectangles. I 4 II 7 6+x.. 3. What equation represents the perimeter of rectangle I? A. 0+x B. 0+x C. 0+x Write an expression that represents the perimeter of rectangle II. If x is an integer, what are the smallest and largest values x can be? 4. Explain how you got your answer to #3. 5. Suppose the areas of the two rectangles are equal. What is the value of x? What is the perimeter of each rectangle? What is the area of each rectangle? 6-x Find the perimeter of each regular polygon. 6. regular hexagon with sides 8.5 millimeters long 7. regular decagon with sides.5 inches long 8. regular heptagon with sides 0.75 feet long 9. regular -gon with sides 3.5 yards long 0. regular 5-gon with sides 6 inches long.. Mary needs to cut a piece of glass for her table. The table is in the shape of a regular hexagon. The glass should measure ft. on each side. What is the perimeter of the piece of glass? A. ft. B. 9 ft. C. 8 ft. D. 7.5 ft. How would you specifically describe the change in perimeter of a triangle if all its side lengths are multiplied by 4?

6 Activity -4: Area of Rectangles Use either the STAAR formula chart to help answer the following problems. Show all work on separate paper including three steps for each problem: write the correct formula, fill in the numbers for the variables, and then solve the equation. Game Shape Dimensions Perimeter Area. Racquetball Rectangle w = ft. l = 40 ft. A = 800 sq. ft.. NCAA basketball Rectangle w = 50 ft. l = 94 ft. 3. Ice hockey Rectangle w = 85 ft. l = ft. P = 570 ft. 4. Volleyball Rectangle w = ft. l = 60 ft. A = 800 sq. ft. 5. Lacrosse Rectangle w = 80 ft. l = 330 ft. 6. NCAA soccer Rectangle w = 5 ft. l = 360 ft. 7. Football Rectangle w = 60 ft. 8. Tennis Rectangle 9. Baseball infield diamond l = 360 ft. w = 36 ft. l = 78 ft. Square s = ft. A = 800 sq. ft. Bloom s Nursery designed a plan for Mrs. Johnsen s flower bed, as shown in the shaded part of the grid below. 0. Each square on the grid represents 5 square feet. What will be the approximate area of the flower bed? A. 00 ft. B. 80 ft. C. 0 ft. D. 6 ft. Mrs. Jones wants to paint a wall but not the door on the wall. 5 ft.. Door: 3 ft by 7 ft 0 ft. How many square feet of wall does Mrs. Jones need to paint?

7 Activity -5: Area of Parallelograms Formula for the area of a parallelogram: A = bh Example: 4 m 8 m 5 m The height is measured straight up from the base. The height of this parallelogram is 4 m. A = bh A = 8 4 A = 3 m. Find the perimeter and the area of each parallelogram. For the area, show all steps... 8 ft 0 ft 5 m m 6 ft cm 3. cm 5. cm m 6.5 cm 9.3 cm 7.5 cm ft 90 ft.0 m.8 m 00 ft 0.7 m The base of a parallelogram is 0 in. The height is in. more than half the base. Find the area. The height of a parallelogram is 4.5 cm. The base is twice the height. What is the area? The area of a parallelogram is 60 ft. The height is 5 ft. How long is the base? The area of a parallelogram is 75 cm. The base is 5 cm. Find the height. Mr. Mangham wants to figure out how many bags of fertilizer he needs to cover his yard. You know the following: the area of the yard, area each bag of fertilizer can cover, cost of each bag, weight of each bag. How would you determine the number of bags needed?

8 Activity -6: Area of Triangles Formula for the area of a triangle: bh A = or bh (Half of the formula for a parallelogram.) Example: 5 in 6 in The height is measured straight up from the base. The height of this triangle is 5 in. bh A = 6 5 A = A = 5 in. Find the area of each triangle using the formula above. Show all steps on a separate sheet of paper.. 5 mm. 3 mm 7.5 cm 8 cm in. in. 6 4 m 3 3 m in. 4 ft. 7 in. 4 ft km 9. 5 yd. yd. 9 km. km. 3.7 km.

9 Activity -7: Area of Triangles Find the area of each triangle using the formula above. Show all steps on a separate sheet of paper.. 5 mm. 3. yd 8 mm yd in 8 ft. 7 cm 0 cm 9 cm 6 in 7 ft. 5 cm in.5 in 4.5 km in 6 in.4 km A triangular sail has a base of 5 m and a height of 0 m. If canvas costs $8 a square meter, find the cost of canvas to make the sail. A square dinner napkin 8 in. on each side is folded along its diagonal. Find the area of the folded napkin. A shuffleboard court as a large isosceles triangle with b = 6 ft and A = 7 ft. What is the length of the shuffleboard court? If you doubled the height of a triangle, what would happen to the area of the triangle? If you doubled both the base and the height of a triangle, what would happen to the area? The area of ABC is greater than the area of DEF. Which must be true? A. The height of ABC is greater than the height of DEF. B. The perimeter of ABC is greater than the perimeter of DEF. C. The sum of the angles of ABC is greater then the sum of the angles of DEF. D. Base and height: At least one of them is greater on ABC.

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11 Activity -8: Area of Trapezoids A trapezoid is a quadrilateral with only one pair of parallel sides. For determining its area one can start with the formula for a parallelogram: A = bh. However with a trapezoid the top and bottom bases are different lengths. Thus, to find the area average the two bases and then multiply times the height. Formula for the area of a trapezoid: A = ( b + b ) h [ ( ) b + b is just the average of the two bases.] in Example: A = ( b + b ) h 6 in A = ( + 5)6 5 in A = (7) 6 The two bases are always parallel to each other. A = 8 in. Find the area of each trapezoid using the formula above. Show all steps on a separate sheet of paper. Trapezoid A Trapezoid A Trapezoid B. x=4 cm, y=6.5 cm, z= cm. x=4 cm, y=0 cm, z=5 cm 3. x=40 m, y=50 m, z=0 m 4. x=7 ft, y=5 ft, z=7 ft Trapezoid B x z y 5. x=6 in, y=6 in, z=9 in 6. x=4 cm, y=78 cm, z= cm 7. x=.8 m, y=.5 m, z=.5 m 8. z y 3 x= in, y= in, z=9 in 4 4 x 9. Cassie draws the following 4 figures. List the shapes in order of area from greatest to least. 0. What happens to the area of a trapezoid if both bases are tripled?. 8 cm 0 cm 6 cm.5 cm 0 cm What happens to the area of a trapezoid if both bases and the height are all divided by 3? 8 cm 0 cm 5 cm

12 Activity -9: Area of Different Shapes Find the area of each figure. Show all steps.. 6 m m 5 in 4 m m m m 3 m 4 m 6 in 8 in 0 in in 9 cm 7 in 7 in in 0 cm 8 cm 5 cm Find the area of the shaded region in each figure. 6. yard with a sandbox 7. wall with windows 8. sidewalk around pool Yard: 5 ft by 0 ft Sandbox: 6 ft by 7 ft Wall: 8 ft by 6 ft Each window: 5 ft by 4 ft Sidewalk: 30 ft by 30 ft Pool: 7 ft by 7 ft A bedroom is 5 ft long and ft wide. How much will it cost to carpet the room if carpeting costs $ per square yard? ( yd = 3 ft) A rose garden in the city park is rectangular and is 9 m wide. If the area of the rectangle is 44 m, what is the length of the garden? Cindy had a rectangular garden last year with an area of 60 sq. ft. This year the garden is one foot wider and three feet shorter than last year, but it has the same area. What were the dimensions of the garden last year? An average gallon of paint will cover 350 sq. feet of wall or ceiling space. All of your ceilings are 8 feet high. Your living room is feet by 8 feet. Your kitchen is 5 feet by 5 feet and your dining room is feet by 4 feet. How many gallons of paint would you need to give one coat of paint to each wall and ceiling? If paint costs $3 a gallon, what would the total cost be?

13 Activity -0: All Area Ms. Wagner painted the outside of the patio door to her house, as shown below. She did not paint the window or the doorknob.. in by 3 in 7 ft. ft by ft (each square) Which is the closest to the painted area of the door in square feet? 4 ft. A. 3 ft. B. 8 ft. C. 5 ft. D. 8 ft. A pest-controlled company was hired to spray the lawn represented by the shaded region shown below. What was the area in square feet that was sprayed?. 4 ft by 30 ft Gar House 00 feet 00 feet 40 ft by 40 ft A. 9,80 ft. B. 0,000 ft. C. 37,680 ft. D. 7,680 ft. Manny made a rectangular garden in his backyard. The garden was 4 feet long and 0 feet wide. Manny used 3 of the garden space to grow vegetables. He built a 3 foot high fence around the garden to keep his dog out of the garden. Determine which of the following questions could NOT be answered with the information provided. A. What is the perimeter of the garden? B. What was the total area of the garden? C. What was the volume of dirt in the garden? D. What was the area of space used for growing vegetables? A farmer knows the length and width of his rectangular pasture. He also knows how many pounds of fertilizer to spread per square yard. What additional information does the farmer need to know in order to determine the number of bags of fertilizer he should buy? A. The type of grass in the pasture B. The number of bags of fertilizer his truck will hold C. The price of each bag of fertilizer D. The number of pounds of fertilizer in each bag An equilateral triangle is divided into 4 congruent equilateral triangles. What method can be used to find the area of the larger equilateral triangle, given the area of one of the smaller triangles? A. Multiply the area of the larger equilateral triangle by 4 B. Multiply the area of one congruent equilateral triangle by 4 C. Subtract the area of one congruent triangle from the area of the larger equilateral triangle D. Add the area of the larger equilateral triangle to the areas of the 4 congruent equilateral triangles

14 Activity -: Area and Perimeter Use graph paper for all drawings and all work.. Draw a figure whose perimeter is 4 units.. Draw a different figure whose perimeter is also 4 units. 3. Draw a figure whose area is 4 square units. 4. Draw a different figure whose area is also 4 square units. 5. Make up a real world word problem in which you need to find the perimeter of any quadrilateral. 6. Make up a real world word problem in which you need to find the area of any quadrilateral. 7. Can two different figures have the same area but different perimeters? Explain your answer. Your dog, Benji, needs a new play area. You are in charge of building a fence around the dog s play area so that he can t run away. You are given 80 feet of fencing to build your play area. Build two different play areas that you think would be suitable for a dog using all of the fencing. For each of your SCALE drawings: 8. Calculate the perimeter 9. Calculate the area 0. Explain why/how you chose the shape for each play area PART 9. 7 in. The perimeter of the rectangle is 6 in. Find the length of each side. 0. Amanda bought 40 meters of fencing to make an enclosure for her dog, Sushi. If Amanda expects a rectangular enclosure, what is the largest area it can have? Explain your answer.. The width of a rectangle is 4.5 inches and its perimeter is 3 inches. What is the length of the rectangle? PART 3. The club house is a rectangle that is 5 feet by 40 feet in size. The officers voted to put a 6-foot sidewalk all around the building, leaving a -foot space for plants between the building and the sidewalk. Give the perimeter of the outer edge of the sidewalk and the area of the sidewalk itself. 3. What is the area of each black and white piece if the whole square measures 0 cm on each side? What percent of the area of the large square is the small shaded square?

15 Activity -POW (Draw a Picture): Hot Tubs Hot tubs and in-ground swimming pools are sometimes surrounded by borders of tiles. You have a square hot tub with sides of length b feet. Your tub is surrounded by a border of square tiles. Each border tile measures foot on each side. How many -foot square tiles will be needed for the square border of the hot tub that has edge length of b feet? Draw separate pictures in which you group the border tiles in different, logical arrangements and then express the total number of tiles needed with equivalent expressions (one for each picture you have drawn). This problem tells you specifically to draw pictures, so that must be the strategy! Luckily for you, the pictures have already been drawn for you on the other side. Your job is to look specifically at how the tiles have been grouped to come up with an appropriate expression using numbers, variables, and operations. Hint: If you simplified all your expressions, you would get the same answer every time. THE HOT TUB b b

16 Four segments + Four corners Two long segments + Two short segments Four segments Four corners (since they are covered twice) All shading Inside shading Can you think of any more logical configurations? If so, draw them and determine their formulas. Four segments

17 Activity -: The Royal Rule In the Kingdom of Squareless, everyone was required to give a parcel of land to the Queen. This land had to be no smaller that that which could be enclosed by 8 meters of fence, and no side could be less than meter long. Most people simply gave the Queen a square plot of land that was seven meters on each side. Peasant Mangham, being the clever one that he was, wanted to give the Queen as little land as possible and he was about to comply with the rule by only giving her 3 square meters of land. He was sentenced to life in prison for trying to outsmart the Queen. Peasant Mangham then asked the Queen if his sentence could be suspended if he could truly amaze the Queen. The Queen agreed. Peasant Mangham showed the Queen the following land (each square is one meter on each side). He asked the Queen how much fence it would take to fence the area. Peasant Mangham then told the Queen he could increase the area by 50% and still use the same amount of fence. The Queen was puzzled and said Peasant Mangham could have his freedom is he could explain how this worked. Use graph paper and notebook paper to answer the following.. Draw a model of the plot of land most people gave the Queen.. Draw a model of the plot of land that Peasant Mangham gave the Queen. 3. Draw the shape made by Peasant Mangham on this page. How many more squares can you enclose and not change the perimeter? Will Peasant Mangham be released from prison? 4. Draw a similar shape to the one above that has a perimeter of 8. How can you increase the area by 75% without changing the perimeter? 5. Draw a shape that has a perimeter of 4. Double the area without changing the perimeter. 6. Draw a shape that has a perimeter of 0. Show how the area can more than double without changing the perimeter. 7. Suppose the Queen wanted you to fence in a rectangular space that had a perimeter of 8 meters. What are the possible dimensions? Give at least 5 examples. 8. What has to be true of each of the pairs of dimensions that you find? 9. Suppose the Queen wants you to fence in a rectangular space that had an area of 8 square meters. What are the possible rectangular dimensions? Give at least 5 examples. 0. What has to be true about each of the pairs of dimensions that you find?

18 Using your results of the possible areas that can be enclosed by 8 meters of fence, answer the following questions.. What is the smallest land area that can be submitted?. 3. What seems to be true of the shape of the land with smaller areas? Is there an even smaller land area that fits the Queen s request? 4. What is the largest land that can be submitted? 5. What seems to be true about the shape of the land with larger areas? 6. Is there an even larger area that fits the Queen s request? 7. What conclusions can be drawn from the above answers? After Peasant Mangham managed to successfully escape being imprisoned by the Queen, he decided to issue his own challenge: Your majesty, if this is really the kingdom of Squareless, who do you insist that all of the parcels of land be squares or rectangles? If I simply give you 8 meters of fencing and no restrictions, what would be the greatest land area you could enclose? I challenge you to break out of your mold! If the Queen hires you to tackle this challenge, what will you answer be? Prove it! 8.