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Area: Parallelograms

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  1. Area: Parallelograms PRE-ALGEBRA LESSON 10-1 It takes 126 ft of fence to enclose a field that is twice as long as it is wide. What is the area of the field? 882 ft2 10-1

  2. Area: Parallelograms PRE-ALGEBRA LESSON 10-1 (For help, go to Lesson 3-4.) Use A = w and find the third value. 1.A = 54 in.2, w = 6 in. 2. = 35 m, w = 7 m 3.A = 25 cm2, = 2.5 cm 4. = 7.2 ft, w = 7.2 ft Check Skills You’ll Need 10-1

  3. Area: Parallelograms PRE-ALGEBRA LESSON 10-1 Solutions 1.A = w2.A = w 54 = 6 A = 35 • 7 = 9 in. A = 245 m2 3.A = w4.A = w 25 = 2.5wA = 7.2 • 7.2 w = 10 cm A = 51.84 ft2 10-1

  4. Area: Parallelograms PRE-ALGEBRA LESSON 10-1 Find the area of the rectangle. Step 1Change the units so that they are the same. 150 cm = 1.5 m Change 150 centimeters to meters. Step 2 Find the area. A = bhUse the formula for area of a rectangle. = (4)(1.5) Replace b and h with the dimensions 4 and 1.5. = 6 Simplify. The area of the rectangle is 6 m2. Quick Check 10-1

  5. A = bharea formula A = bh = (8)(2) Substitute. = (2.5)(6) = 16 Simplify. = 15 Area: Parallelograms PRE-ALGEBRA LESSON 10-1 Find the area of each parallelogram. a. b. The area is 16 m2. The area is 15 in.2. Quick Check 10-1

  6. Area: Parallelograms PRE-ALGEBRA LESSON 10-1 Find each area. 1. rectangle ABFE 2. parallelogram ACFD 3. a rectangle with a base of 50 cm and a height of 5 cm 40 cm2 48 cm2 250 cm2 10-1

  7. 12 12 24 ft2 Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 Find the area of a rectangle that is 3 ft wide and twice as high. 10-2

  8. 1 2 1 2 1 2 1 2 1 2 Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 (For help, go to Lesson 5-4.) Find each product. 1. • 16 2. • 14 • 6 3. • 5 • 15 4. • 2 • 8 Check Skills You’ll Need 10-2

  9. 1 2 1 2 1 2 1 2 1 2 1 2 1 2 5 2 1 2 Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 Solutions 1. • 16 2. • 14 • 6 • 168 = 8 • 147 • 6 7 • 6 = 42 3. • 5 • 15 4. • 2 • 8 • 75 = 37 • • 8 • • 82 5 • 2 = 10 1 21 1 21 1 21 5 21 10-2

  10. A = bhUse the formula for area of a triangle. = • 13 • 6 Replace b with 13 and h with 6. 1 2 1 2 Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 Find the area of the triangle. = 39 Simplify. Quick Check The area is 39 in.2. 10-2

  11. Area of triangle Area of rectangle A = bh A = bh = • 45 • 20 = 45 • 30 = 450 = 1,350 1 2 1 2 Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 Find the area of the figure. Add to find the total: 450 + 1,350 = 1,800. Quick Check The area of the figure is 1,800 cm2. 10-2

  12. A = h(b1 + b2) A = • 3(31 + 40) Replace h with 3, b1 with 31, and b2 with 40. Use the formula for the area of a trapezoid. = • 3(71) Simplify. = • 213 1 2 1 2 1 2 1 2 Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 Suppose that, through the years, a layer of silt and mud settled in the bottom of the Erie Canal. Below is the resulting cross section of the canal. Find the area of the trapezoidal cross section. = 106.5 Quick Check The area of the cross section is 106.5 ft2. 10-2

  13. Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10-2 Find each area. 1. trapezoid PQRU2. triangle PTU 3. triangle QRS4. trapezoid PQSU 192 ft2 20 ft2 28 ft2 164 ft2 10-2

  14. Area: Circles PRE-ALGEBRA LESSON 10-3 Explain how this pattern works. (15,873  7)  2 = (222,222) (15,873  7)  3 = (333,333) (15,873  7)  4 = (444,444) because (15,873  7)  1 = (111,111) 10-3

  15. Area: Circles PRE-ALGEBRA LESSON 10-3 (For help, go to Lesson 4-7.) Simplify each expression. 1. 3.14 • 422. 3.14 • 52 3. 3.14 • 924. 3.14 • 0.52 Check Skills You’ll Need 10-3

  16. Area: Circles PRE-ALGEBRA LESSON 10-3 Solutions 1. 3.14 • 422. 3.14 • 52 = 3.14 • 4 • 4 = 3.14 • 5 • 5 = 50.24 = 78.5 3. 3.14 • 924. 3.14 • 0.52 = 3.14 • 9 • 9 = 3.14 • 0.5 • 0.5 = 254.34 = 0.785 10-3

  17. A = r 2 1 2 = (10)2r = d; r = 10 = 100 Simplify. The area is 100 in.2. Area: Circles PRE-ALGEBRA LESSON 10-3 Find the exact area of a circle with diameter 20 in. Quick Check 10-3

  18. A = r 2 = (50)2r = d; r = 50 = 2,500 exact area 1 2 (2,500)(3.14) Use 3.14 for . Area: Circles PRE-ALGEBRA LESSON 10-3 A TV station’s weather radar can detect precipitation in a circular region having a diameter of 100 mi. Find the area of the region. = 7,850 approximate area The area of the region is about 7,850 mi2. Quick Check 10-3

  19. Area of region that is one fourth of a circle: area of circle = r 2 area of quarter circle = r 2 A (3.14)(15)2 Replace with 3.14 and r with 15. = 176.625 ft2 1 4 1 4 Area: Circles PRE-ALGEBRA LESSON 10-3 A pound of grass seed covers approximately 675 ft2. Find the area of the lawn below. Then find the number of bags of grass seed you need to buy to cover the lawn. Grass seed comes in 3-lb bags. 10-3

  20. 1,302 ÷ 675 1.93 Divide to find the number of pounds of seed. Area: Circles PRE-ALGEBRA LESSON 10-3 (continued) Area of region that is a rectangle: area of rectangle = bh A = 45 • 25 Replace b with 45 and h with 25. = 1,125 ft2 The area of the lawn is about 177 ft2 + 1,125 ft2 = 1,302 ft2. You need to buy one 3-lb bag of grass seed. Quick Check 10-3

  21. 256 in.2 Area: Circles PRE-ALGEBRA LESSON 10-3 Find the area. 1. Find the exact area of a circle with diameter 32 in. 2. A 5-ft-diameter round table is in a 12 ft-by-15 ft room. a. What is the area covered by the table? Round to the nearest unit. b. What is the area of the rest of the room? Round to the nearest unit. 20 ft2 160 ft2 10-3

  22. Space Figures PRE-ALGEBRA LESSON 10-4 Find the area of a square 15.7 mm on each side. 246.49 mm2 10-4

  23. Space Figures PRE-ALGEBRA LESSON 10-4 (For help, go to Lesson 9-3.) Judging by appearance, classify each polygon. 1. 2. 3. 4. Check Skills You’ll Need 10-4

  24. Space Figures PRE-ALGEBRA LESSON 10-4 Solutions 1. triangle 2. square 3. rectangle 4. hexagon 10-4

  25. Space Figures PRE-ALGEBRA LESSON 10-4 For each figure describe the bases and name the figure. a. b. The bases are circles. The bases are rectangles. The figure is a cylinder. The figure is a rectangular prism. Quick Check 10-4

  26. Space Figures PRE-ALGEBRA LESSON 10-4 Name the space figure you can form from each net. a. b. With two hexagonal bases and rectangular sides, you can form a hexagonal prism. With a rectangular base and triangular sides, you can form a rectangular pyramid. Quick Check 10-4

  27. Space Figures PRE-ALGEBRA LESSON 10-4 Describe the base(s) and name each solid. 1. 2. 3. Name the solid you can form from this net. circular base; cone triangular bases; triangular prism octagonal pyramid 10-4

  28. Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 A prism has 7 faces and 10 vertices. How many edges does the prism have? 15 10-5

  29. Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 (For help, go to Lesson 9-6.) Find the circumference of each circle with the given radius or diameter. 1.r = 5 in. 2. r = 4.2 cm 3.d = 8 ft 4.d = 6.8 in. Check Skills You’ll Need 10-5

  30. Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 Solutions 1.C = 2 r2.C = 2 r C = 2(3.14)(5 in.) C = 2(3.14)(4.2 cm) C = 31.4 in. C = 26.4 cm 3. C = d4.C = d C = (3.14)(8 ft) C = (3.14)(6.8 in.) C = 25.1 ft C = 21.3 in. 10-5

  31. Find the area of each rectangle in the net. Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 Find the surface area of the rectangular prism using a net. Draw and label a net. 60 + 60 + 150 + 90 + 150 + 90 = 600 Add the areas. Quick Check The surface area is 600 cm2. 10-5

  32. Step 1 Find the lateral area. L.A. = phUse the formula for lateral area. = (5 + 6 + 5 + 6)20 p = 5 + 6 + 5 + 6 and h = 20 = 440 Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 Find the surface area of the rectangular prism. Step 2 Find the surface area. S.A. = L.A. + 2B = 440 + 2(5 • 6) L.A. = 440 and B = 5 • 6 = 440 + 60 = 500 Quick Check The surface area of the rectangular prism is 500 in.2. 10-5

  33. Step 1 Find the lateral area. L.A. = 2 rhUse the formula for lateral area. 2(3.14)(8)(15) 754 Step 2 Find the surface area. S.A. = L.A. + 2B Use the formula for surface area. = L.A. + 2( r 2) 754 + 2(3.14)(8)2 1,156 Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 Find the surface area of the cylindrical water tank. Quick Check The surface area of the water tank is about 1,156 ft2. 10-5

  34. Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10-5 Find the surface area of each figure rounded to the nearest whole unit. 1. triangular prism with base perimeter 24 cm, base area 24 cm2, and height 15 cm 2. rectangular prism with base perimeter 30 cm, base area 50 cm2, and height 150 cm 3. cylindrical candle with radius 2 cm and height 16 cm 408 cm2 4,600 cm2 about 226 cm2 10-5

  35. Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 Give the number of faces, edges, and vertices of a rectangular prism. 6 faces, 12 edges, 8 vertices 10-6

  36. 2 3 1 2 3 4 2 5 1 6 1 3 5 8 1 7 Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 (For help, go to Lesson 5-4.) Use the Order of Operations to simplify each expression. 1. (9 ) + (8 ) 2. (12 ) + (15 ) 3. (24 ) + (3 ) 4. (32 ) + (14 ) Check Skills You’ll Need 10-6

  37. 1 21 153 123 142 Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 Solutions 1. • 93 + • 842. • + • = 6 + 4 = 9 + 6 = 10 = 15 3. • 244 + • 314. • 324 + • = 4 + = 20 + 2 = 5 = 22 2 31 3 41 2 51 1 61 1 71 1 31 5 81 10-6

  38. Step 1L.A. = p Use the formula for lateral area. = • 20 • 8 = 80p = 4(5) and = 8. 1 2 1 2 Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 Find the surface area of the square pyramid. Step 2 S.A. = L.A. + B = 80 + 52 Lateral area = 80 and B = 52. = 80 + 25 = 105 Quick Check The surface area of the pyramid is 105 m2. 10-6

  39. Step 1L.A. = r Use the formula for lateral area. 3.14(3)(7)r = 3 and = 7. = 65.94 Step 2 S.A. = L.A. + B Use the formula for surface area. 65.94 + 3.14(3)2L.A. 65.94 and B = (3)2. = 65.94 + 28.26 = 94.2 Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 Find the surface area of the cone. Quick Check The surface area of the cone is about 94 m2. 10-6

  40. S.A. = 4 r 2Use the formula for surface area. 4(3.14)(3,963)2r 3,963 197,259,000 Round to nearest 1,000. Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 Earth has an average radius of 3,963 mi. What is Earth’s approximate surface area to the nearest 1,000 mi2? Assume that Earth is a sphere. = 197,259,434.64 Multiply. Quick Check The surface area of Earth is about 197,259,000 mi2. 10-6

  41. Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10-6 Find the surface area of each space figure. Round to the nearest whole unit. 1. a square pyramid with base edge 80 m and slant height 100 m 2. a cone with slant height 22 cm and radius 7 cm 3. a sphere with radius 12 cm 22,400 m2 about 637 cm2 about 1,809 cm2 10-6

  42. Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7 Use graph paper. Design and draw a diagram to determine which has the greater area—a square with sides 10 cm or a circle with a diameter 10 cm? Draw a circle within the square to prove that the square has the greater area. 10-7

  43. Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7 (For help, go to Lesson 10-3.) Find the area of each circle. 1. 2. 3. Check Skills You’ll Need 10-7

  44. 3.r = d 1. A = r2 2. A = r2 1 2 1 2 Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7 Solutions = (82)( ) = (122)( ) = 64(3.14) = 144(3.14) 201 cm2 452.2 cm2 = (20) = 10 A = r2 = (102)( ) = 100(3.14) 314 cm2 10-7

  45. 1 2 = 63 • 20B = • 9 • 14 = 63 cm2 Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7 Find the volume of the triangular prism. V = Bh Use the formula for volume. = 1,260Simplify. The volume is 1,260 cm3. Quick Check 10-7

  46. V = r 2h B = r 2 3.14 • 3.42 • 16Replace r with 3.4, and h with 16. Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7 Find the volume of the juice can, to the nearest cubic centimeter. V = Bh Use the formula for volume. = 580.7744 Simplify. Quick Check The volume is about 581 cm3. 10-7

  47. Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10-7 Find the volume of each space figure. 1. rectangular prism with base 12 m by 14 m and height 50 m 2. cylindrical pool with diameter 24 ft and height 4 ft 3. right triangular prism with base legs 8 cm and 10 cm and height 20 cm 8,400 m3 about 1,808.64 ft3 800 cm3 10-7

  48. 12 13 23 23 ft2 Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8 A roll of wallpaper is 24 in. wide. You used all but 4 ft of one roll. Another roll has 7 ft of wallpaper on it. What is the total area you can cover with the remaining wallpaper? 10-8

  49. Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8 (For help, go to Lesson 9-3.) Draw each figure described below. 1. a rectangle with small squares drawn in each corner 2. a rectangle divided into eight congruent rectangles 3. two parallelograms that have different shapes but the same perimeter Check Skills You’ll Need 10-8

  50. Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10-8 Solutions Answers may vary. Samples: 1.2. 3. 10-8