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 ft 2 10 -1
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 cm 2, = 2. 5 cm 4. = 7. 2 ft, w = 7. 2 ft Check Skills You’ll Need 10 -1
Area: Parallelograms PRE-ALGEBRA LESSON 10 -1 Solutions 1. A = w 54 = 6 = 9 in. 3. A = w 25 = 2. 5 w w = 10 cm 2. A = w A = 35 • 7 A = 245 m 2 4. A = w A = 7. 2 • 7. 2 A = 51. 84 ft 2 10 -1
Area: Parallelograms PRE-ALGEBRA LESSON 10 -1 Find the area of the rectangle. Step 1 Change the units so that they are the same. 150 cm = 1. 5 m Change 150 centimeters to meters. Step 2 Find the area. A = bh Use 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 m 2. 10 -1 Quick Check
Area: Parallelograms PRE-ALGEBRA LESSON 10 -1 Find the area of each parallelogram. a. b. A = bh area formula A = bh = (8)(2) Substitute. = (2. 5)(6) = 16 Simplify. = 15 The area is 16 m 2. The area is 15 in. 2. 10 -1 Quick Check
Area: Parallelograms PRE-ALGEBRA LESSON 10 -1 Find each area. 1. rectangle ABFE 40 cm 2 2. parallelogram ACFD 48 cm 2 3. a rectangle with a base of 50 cm and a height of 5 cm 250 cm 2 10 -1
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10 -2 1 2 Find the area of a rectangle that is 3 ft wide and twice as high. 1 24 ft 2 2 10 -2
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10 -2 (For help, go to Lesson 5 -4. ) Find each product. 1 1. • 16 2 1 3. 2 • 5 • 15 1 2 1 1 4. • 2 • 8 2 2 2. • 14 • 6 Check Skills You’ll Need 10 -2
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10 -2 Solutions 1. 1 • 16 2 1 • 168 = 8 21 3. 1 • 5 • 15 2 1 1 • 75 = 37 2 2 1 2. • 14 • 6 2 1 • 147 • 6 21 7 • 6 = 42 1 1 4. • 2 • 8 2 1 2 5 • 2 • 8 2 1 5 • • 82 21 21 5 • 2 = 10 10 -2
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10 -2 Find the area of the triangle. 1 2 Use the formula for area of a triangle. = • 13 • 6 1 2 Replace b with 13 and h with 6. = 39 Simplify. A = bh Quick Check The area is 39 in. 2. 10 -2
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10 -2 Find the area of the figure. Area of triangle Area of rectangle 1 2 1 = • 45 • 20 2 A = bh = 45 • 30 = 450 = 1, 350 Add to find the total: 450 + 1, 350 = 1, 800. The area of the figure is 1, 800 cm 2. 10 -2 Quick Check
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. 1 2 1 A = 2 1 = 2 A = h(b 1 + b 2) Use the formula for the area of a trapezoid. • 3(31 + 40) Replace h with 3, b 1 with 31, and b 2 with 40. • 3(71) Simplify. • 213 = 106. 5 The area of the cross section is 106. 5 ft 2. 10 -2 Quick Check
Area: Triangles and Trapezoids PRE-ALGEBRA LESSON 10 -2 Find each area. 1. trapezoid PQRU 192 ft 2 2. triangle PTU 3. triangle QRS 28 ft 2 4. trapezoid PQSU 164 ft 2 20 ft 2 10 -2
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
Area: Circles PRE-ALGEBRA LESSON 10 -3 (For help, go to Lesson 4 -7. ) Simplify each expression. 1. 3. 14 • 42 3. 3. 14 • 92 2. 3. 14 • 52 4. 3. 14 • 0. 52 Check Skills You’ll Need 10 -3
Area: Circles PRE-ALGEBRA LESSON 10 -3 Solutions 1. 3. 14 • 42 = 3. 14 • 4 = 50. 24 3. 3. 14 • 92 = 3. 14 • 9 = 254. 34 2. 3. 14 • 52 = 3. 14 • 5 = 78. 5 4. 3. 14 • 0. 52 = 3. 14 • 0. 5 = 0. 785 10 -3
Area: Circles PRE-ALGEBRA LESSON 10 -3 Find the exact area of a circle with diameter 20 in. A = r 2 1 2 = (10)2 r = d; r = 100 Simplify. The area is 100 in. 2. Quick Check 10 -3
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. A = r 2 1 2 = (50)2 r = d; r = 50 = 2, 500 exact area (2, 500)(3. 14) Use 3. 14 for . = 7, 850 approximate area The area of the region is about 7, 850 mi 2. Quick Check 10 -3
Area: Circles PRE-ALGEBRA LESSON 10 -3 A pound of grass seed covers approximately 675 ft 2. 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. Area of region that is one fourth of a circle: area of circle = r 2 1 area of quarter circle = r 2 4 1 A (3. 14)(15)2 4 Replace with 3. 14 and r with 15. = 176. 625 ft 2 10 -3
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 ft 2 The area of the lawn is about 177 ft 2 + 1, 125 ft 2 = 1, 302 ft 2. 1, 302 ÷ 675 1. 93 Divide to find the number of pounds of seed. You need to buy one 3 -lb bag of grass seed. 10 -3 Quick Check
Area: Circles PRE-ALGEBRA LESSON 10 -3 Find the area. 1. Find the exact area of a circle with diameter 32 in. 256 in. 2 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. 20 ft 2 b. What is the area of the rest of the room? Round to the nearest unit. 160 ft 2 10 -3
Space Figures PRE-ALGEBRA LESSON 10 -4 Find the area of a square 15. 7 mm on each side. 246. 49 mm 2 10 -4
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
Space Figures PRE-ALGEBRA LESSON 10 -4 Solutions 1. triangle 3. rectangle 2. square 4. hexagon 10 -4
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
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
Space Figures PRE-ALGEBRA LESSON 10 -4 Describe the base(s) and name each solid. 1. 2. circular base; cone triangular bases; triangular prism 3. Name the solid you can form from this net. octagonal pyramid 10 -4
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
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
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10 -5 Solutions 1. C = 2 r C = 2(3. 14)(5 in. ) C = 31. 4 in. 3. C = d C = (3. 14)(8 ft) C = 25. 1 ft 2. C = 2 r C = 2(3. 14)(4. 2 cm) C = 26. 4 cm 4. C = d C = (3. 14)(6. 8 in. ) C = 21. 3 in. 10 -5
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. Find the area of each rectangle in the net. 60 + 150 + 90 = 600 Add the areas. Quick Check The surface area is 600 cm 2. 10 -5
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10 -5 Find the surface area of the rectangular prism. Step 1 Find the lateral area. L. A. = ph Use the formula for lateral area. = (5 + 6 + 5 + 6)20 p = 5 + 6 + 5 + 6 and h = 20 = 440 Step 2 Find the surface area. S. A. = L. A. + 2 B = 440 + 2(5 • 6) L. A. = 440 and B = 5 • 6 = 440 + 60 = 500 The surface area of the rectangular prism is 500 in. 2. 10 -5 Quick Check
Surface Area: Prisms and Cylinders PRE-ALGEBRA LESSON 10 -5 Find the surface area of the cylindrical water tank. Step 1 Find the lateral area. L. A. = 2 rh Use the formula for lateral area. 2(3. 14)(8)(15) 754 Step 2 Find the surface area. S. A. = L. A. + 2 B Use the formula for surface area. = L. A. + 2( r 2) 754 + 2(3. 14)(8)2 1, 156 The surface area of the water tank is about 1, 156 ft 2. 10 -5 Quick Check
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 cm 2, and height 15 cm 408 cm 2 2. rectangular prism with base perimeter 30 cm, base area 50 cm 2, and height 150 cm 4, 600 cm 2 3. cylindrical candle with radius 2 cm and height 16 cm about 226 cm 2 10 -5
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
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. 2 1 3 2 1. (9 ) + (8 ) 2. (12 ) + (15 ) 3 2 4 5 1 1 3. (24 ) + (3 ) 6 3 5 1 4. (32 ) + (14 ) 8 7 Check Skills You’ll Need 10 -6
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -6 Solutions 1 1. 21 • 93 + • 84 3 21 = 6 + 4 = 10 1 1 3. • 244 + • 31 61 31 = 4 + = 5 2 2. 31 123 • + • 153 1 4 5 = 9 + 6 = 15 1 4. 51 • 324 + • 142 1 8 = 20 + 2 = 22 10 -6 7
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -6 Find the surface area of the square pyramid. 1 2 Use the formula for lateral area. 1 2 p = 4(5) and = 8. Step 1 L. A. = p = • 20 • 8 = 80 Step 2 S. A. = L. A. + B = 80 + 52 = 80 + 25 = 105 Lateral area = 80 and B = 52. The surface area of the pyramid is 105 m 2. 10 -6 Quick Check
Surface Area: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -6 Find the surface area of the cone. Step 1 L. A. = r Use the formula for lateral area. 3. 14(3)(7) = 65. 94 r = 3 and = 7. Step 2 S. A. = L. A. + B Use the formula for surface area. 65. 94 + 3. 14(3)2 L. A. 65. 94 and B = (3)2. = 65. 94 + 28. 26 = 94. 2 Quick Check The surface area of the cone is about 94 m 2. 10 -6
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 mi 2? Assume that Earth is a sphere. S. A. = 4 r 2 Use the formula for surface area. 4(3. 14)(3, 963)2 r 3, 963 = 197, 259, 434. 64 Multiply. 197, 259, 000 Round to nearest 1, 000. The surface area of Earth is about 197, 259, 000 mi 2. 10 -6 Quick Check
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 22, 400 m 2 2. a cone with slant height 22 cm and radius 7 cm about 637 cm 2 3. a sphere with radius 12 cm about 1, 809 cm 2 10 -6
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
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
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10 -7 Solutions 1. A = r 2 = (82)( ) = 64(3. 14) 201 cm 2 2. A = r 2 = (122)( ) = 144(3. 14) 452. 2 cm 2 10 -7 1 3. r = d 2 1 = (20) 2 = 10 A = r 2 = (102)( ) = 100(3. 14) 314 cm 2
Volume: Prisms and Cylinders PRE-ALGEBRA LESSON 10 -7 Find the volume of the triangular prism. V = Bh Use the formula for volume. 1 2 = 63 • 20 B = • 9 • 14 = 63 cm 2 = 1, 260 Simplify. The volume is 1, 260 cm 3. Quick Check 10 -7
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. V = r 2 h B = r 2 3. 14 • 3. 42 • 16 = 580. 7744 Replace r with 3. 4, and h with 16. Simplify. The volume is about 581 cm 3. 10 -7 Quick Check
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 8, 400 m 3 2. cylindrical pool with diameter 24 ft and height 4 ft about 1, 808. 64 ft 3 3. right triangular prism with base legs 8 cm and 10 cm and height 20 cm 800 cm 3 10 -7
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10 -8 1 A roll of wallpaper is 24 in. wide. You used all but 4 ft of one roll. 2 1 Another roll has 7 ft of wallpaper on it. What is the total area you can 3 cover with the remaining wallpaper? 2 23 ft 2 3 10 -8
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
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10 -8 Solutions Answers may vary. Samples: 1. 2. 3. 10 -8
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10 -8 A can company rolls rectangular pieces of metal that measure 8 in. by 10 in. to make the sides of cans. Which height, 8 in. or 10 in. , will make the can with the greater volume? Build two cans using 8 in. -by-10 in. pieces of paper. You do not need to make the bases, just the sides. not to scale 10 -8
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10 -8 (continued) not to scale Measure your models to find approximate radii. Radius of 10 -in. high can 1. 3 in. Radius of 8 -in. high can 1. 6 in. Find the volumes. V = r 2 h (3. 14)(1. 32)(10) (3. 14)(1. 62)(8) 53. 1 in. 3 64. 3 in. 3 The volume of the can with height 8 in. is greater. 10 -8 Quick Check
Problem Solving Strategy: Make a Model PRE-ALGEBRA LESSON 10 -8 Solve. 1. You cut square corners from a piece of cardboard that has dimensions 32 cm by 40 cm. You then fold the cardboard to create a box with no lid. To the nearest centimeter, what are the dimensions of the box that will have the greatest volume? 20 cm by 28 cm by 6 cm 10 -8
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -9 Multiply. Write each answer in simplest form. 17 1 3 a. 3 2 4 1 8 b. 1 2 1 c. 2 7 5 8 1 6 2 7 16 2 35 10 -9
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -9 (For help, go to Lesson 5 -4. ) Multiply. 1 1. (3. 14)(2)2(5) 3 4 3. (3. 14)(2)3 3 1 3 4 4. 3 (3. 14)(0. 5)3 2. (4)2(6) Check Skills You’ll Need 10 -9
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -9 Solutions 1 3 1 1. (3. 14)(22)(5) = (3. 14)(4)(5) 3 1 1 2. (42)(6) = (16)(6) 3 3 1 = (62. 8) = (96) 3 = 20. 93 = 32 4 1 4. (3. 14)(0. 5)3 = (3. 14)(0. 125) 3 4 3 4 = (25. 12) 3 3. (3. 14)(23) = (3. 14)(8) 3 4 = (0. 3925) 3 = 33. 493 = 0. 523 10 -9
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -9 Find the volume of the cone. 1 3 V = Bh 1 3 V = r 2 h 1 3 (3. 14)(2)2(12) = 50. 24 Use the formula for volume. B = r 2 Replace r with 2 and h with 12. Simplify. The volume of the cone is about 50 in. 3. 10 -9 Quick Check
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -9 Find the volume of the square pyramid. 1 3 1 V = s 2 h 3 1 = (8)2(12) 3 V = Bh = 256 Use the formula for volume. B = s 2 Replace s with 8 and h with 12. Simplify. The volume of the pyramid is 256 in. 3. 10 -9 Quick Check
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -9 Earth has an average radius of 3, 963 mi. What is Earth’s approximate volume to the nearest 1, 000 mi 3? Assume that Earth is a sphere. 4 3 Use the volume formula. (3. 14)(3, 963)3 4 3 Replace r with 3, 963. 260, 579, 713, 159 Simplify. V = r 3 The volume of the Earth is about 260, 580, 000 mi 3. Quick Check 10 -9
Volume: Pyramids, Cones, and Spheres PRE-ALGEBRA LESSON 10 -9 Find the volume of each space figure to the nearest unit. 1. a cone with diameter 9 cm and height 12 cm about 254 cm 2 2. a square pyramid with base edges 12 m and height 18 m 864 m 3 3. a basketball with diameter 10 in. about 523 in. 3 10 -9
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