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Find the area of the triangle. Check It Out: Example 1B A = 1 2 bh Write the formula. A = 54 The area is 54 in 2 . A = 1 2 (108) Multiply. 24 ft 4 ft 1 2 A = 1 2 ( 4 • 24 ) 1 2 Substitute 4 for b and 24 for h . 1 2
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Find the area of the trapezoid. Additional Example 3: Finding the Area of a Trapezoid A = 1 2 h ( b 1 + b 2 ) Use the formula. A = 53 The area is 53 yd 2 . Multiply. A = 1 2 · 4(26 ) 1 2 Substitute 4 for h , 14 for b 1 , and 12 for b 2 . 1 2 A = 1 2 · 4 ( 14 + 12 ) 1 2
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Lesson Quiz Find the area of each triangle. 1. 3. 84 mi 2 2. 4. Find the area of each trapezoid. 39.9 cm 2 22.5 m 2 113 in 2 3 4
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Additional Example 1A: Finding Areas of Composite Figures Find the area of the polygon. Think: Break the polygon apart into rectangles. Find the area of each rectangle. 1.7 cm 4.9 cm 1.3 cm 2.1 cm
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Additional Example 1A Continued A = l w A = l w A = 4.9 • 1.7 A = 2.1 • 1.3 Write the formula for the area of a rectangle. A = 8.33 A = 2.73 8.33 + 2.73 = 11.06 Add to find the total area. The area of the polygon is 11.06 cm 2 . 1.7 cm 4.9 cm 1.3 cm 2.1 cm
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Think: Break the figure apart into a rectangle and a triangle. Find the area of each polygon. Additional Example 1B: Finding Areas of Composite Figures Find the area of the polygon.
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Additional Example 1B Continued A = l w A = 28 • 24 A = 672 A = 168 672 + 168 = 840 Add to find the total area of the polygon. The area of the polygon is 840 ft 2 . A = b h 1 2 __ A = • 28 • 12 1 2 __
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Lesson Quiz 1. Find the area of the figure shown. 220 units 2 2. Phillip designed a countertop. Use the coordinate grid to find its area. 30 units 2
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Lesson Quiz Find how the perimeter and area of the triangle change when its dimensions change. The perimeter is multiplied by 2, and the area is multiplied by 4; perimeter = 24, area = 24; perimeter = 48, area = 96. Insert Lesson Title Here Course 1 10-4 Comparing Perimeter and Area
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Additional Example 1A: Estimating the Area of a Circle Estimate the area of the circle. Use 3 to approximate pi . A = r 2 Write the formula for area. Replace with 3 and r with 20 . Use the order of operations. Multiply. A ≈ 3 • 20 2 A ≈ 1200 m 2 Course 1 10-5 Area of Circles 19.7 m A ≈ 3 • 400
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Additional Example 1B: Estimating the Area of a Circle Estimate the area of the circle. Use 3 to approximate pi . 28 m A = r 2 Write the formula for area. Replace with 3 and r with 14 . Use the order of operations. Divide. r = d ÷ 2 The length of the radius is half the length of the diameter. Multiply. r = 28 ÷ 2 A ≈ 3 • 14 2 Course 1 10-5 Area of Circles r = 14 A ≈ 3 • 196 A ≈ 588 m 2
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Additional Example 2A: Using the Formula for the Area of a Circle Find the area of the circle. Use for pi . Write the formula to find the area. A = r 2 The length of the diameter is twice the length of the radius. Use the order of operations. A 50.29 ft 2 Divide. 22 7 8 ft r = d ÷ 2 r = 8 ÷ 2 = 4 Replace with and r with 4 . 22 7 __ A • 16 22 7 __ A • ( 4 ) 2 22 7 Course 1 10-5 Area of Circles 352 7
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Lesson Quiz: Part I Estimate the area of each circle. 1. 2. Insert Lesson Title Here 3 km 27 km 2 1200 yd 2 38 yd Course 1 10-5 Area of Circles
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3. 4. Insert Lesson Title Here 4.53 cm 2 1.54 m 2 Lesson Quiz: Part II Find the area of each circle. Use for pi . 22 7 2.4 cm 0.7 m 5. A coater has a diameter of 6 inches. Find the area of the largest cup the coaster can hold. Use 3.14 for pi . 28.26 in 2 Course 1 10-5 Area of Circles
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A polyhedron is a three-dimensional object, or solid figure, with flat surfaces, called faces , that are polygons. When two faces of a three-dimensional figure share a side, they form an edge . On a three-dimensional figure, a point at which three or more edges meet is a vertex (plural: vertices ). Course 1 10-6 Three-Dimensional Figures
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Additional Example 1: Identifying Faces, Edges, and Vertices Identify the number of faces, edges, and vertices on each three-dimensional figure. A. B. 5 faces 8 edges 5 vertices 7 faces 15 edges 10 vertices Course 1 10-6 Three-Dimensional Figures
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A prism is a polyhedron with two congruent, parallel bases , and other faces that are all parallelograms. A prism is named for the shape of its bases. A cylinder also has two congruent, parallel bases, but bases of a cylinder are circular. A cylinder is not a polyhedron because not every surface is a polygon. Course 1 10-6 Three-Dimensional Figures
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A pyramid has one polygon shaped base, and the other faces are triangles that come to a point. A pyramid is named for the shape of its base. A cone has a circular base and a curved surface that comes to a point. A cones is not a polyhedron because not every surface is a polygon. Course 1 10-6 Three-Dimensional Figures
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Lesson Quiz 1. Identify the number of faces, edges, and vertices in the figure shown. Identify the figure described 2. two congruent circular faces connected by a curved surface 3. one flat circular face and a curved lateral surface that comes to a point cylinder 8 faces, 18 edges, and 12 vertices Insert Lesson Title Here cone Course 1 10-6 Three-Dimensional Figures
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Additional Example 1: Finding the Volume of a Rectangular Prism Find the volume of the rectangular prism. V = lwh Write the formula. V = 26 • 11 • 13 l = 26 ; w = 11 ; h = 13 Multiply. V = 3,718 in 3 13 in. 26 in. 11 in. Course 1 10-7 Volume of Prisms
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Additional Example 2A: Finding the Volume of a Triangular Prism Find the volume of each triangular prism. Multiply. V = 10.14 m 3 V = Bh Write the formula. V = ( • 3.9 • 1.3 ) • 4 1 2 __ B = • 3.9 • 1.3 ; h = 4 . 1 2 __ Course 1 10-7 Volume of Prisms
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Lesson Quiz Find the volume of each figure. 1. rectangular prism with length 20 cm, width 15 cm, and height 12 cm 2. triangular prism with a height of 12 cm and a triangular base with base length 7.3 cm and height 3.5 cm 3. Find the volume of the figure shown. Insert Lesson Title Here 3,600 cm 3 153.3 cm 3 38.13 cm 3 Course 1 10-7 Volume of Prisms
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Additional Example 1A: Finding the Volume of a Cylinder Find the volume V of the cylinder to the nearest cubic unit. V = r 2 h Write the formula. Replace with 3.14 , r with 4, and h with 7. Multiply. V 351.68 V 3.14 4 2 7 The volume is about 352 ft 3 . Course 1 10-8 Volume of Cylinders
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Check It Out: Example 1B 8 cm 6 cm V = r 2 h Multiply. V 301.44 8 cm ÷ 2 = 4 cm The volume is about 301 cm 3 . Find the radius. Write the formula. Replace with 3.14 , r with 4, and h with 16. V 3.14 4 2 6 Course 1 10-8 Volume of Cylinders
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Lesson Quiz: Part I Find the volume of each cylinder to the nearest cubic unit. Use 3.14 for . Insert Lesson Title Here cylinder b 1. radius = 9 ft, height = 4 ft 2. radius = 3.2 ft, height = 6 ft <ul><li>3. Which cylinder has a greater volume? </li></ul><ul><ul><li>a. radius 5.6 ft and height 12 ft </li></ul></ul><ul><ul><li>b. radius 9.1 ft and height 6 ft </li></ul></ul>1,560.14 ft 3 193 ft 3 1,017 ft 3 1,181.64 ft 3 Course 1 10-8 Volume of Cylinders
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Lesson Quiz: Part II Insert Lesson Title Here about 396 in 2 4. Jeff’s drum kit has two small drums. The first drum has a radius of 3 in. and a height of 14 in. The other drum has a radius of 4 in. and a height of 12 in. Estimate the volume of each cylinder to the nearest cubic inch. a. First drum b. Second drum about 603 in 2 Course 1 10-8 Volume of Cylinders
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Additional Example 2: Finding the Surface Area of a Pyramid Find the surface area S of the pyramid. S = area of square + 4 (area of triangular face) S = 49 + 4 28 S = 49 + 112 S = 161 The surface area is 161 ft 2 . Substitute. S = s 2 + 4 ( bh ) 1 2 __ S = 7 2 + 4 ( 7 8 ) 1 2 __ Course 1 10-9 Surface Area
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Additional Example 3: Finding the Surface Area of a Cylinder Find the surface area S of the cylinder. Use 3.14 for , and round to the nearest hundredth. S = area of lateral surface + 2 (area of each base) Substitute. S = h (2 r ) + 2 ( r 2 ) S = 7 (2 4 ) + 2 ( 4 2 ) ft Course 1 10-9 Surface Area
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Lesson Quiz Find the surface area of each figure. Use 3.14 for . 1. rectangular prism with base length 6 ft, width 5 ft, and height 7 ft 2. cylinder with radius 3 ft and height 7 ft 3. Find the surface area of the figure shown. Insert Lesson Title Here 214 ft 2 188.4 ft 2 208 ft 2 Course 1 10-9 Surface Area
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You can use the information in the table below to convert one customary unit to another. When you convert units of measure to another, you can multiply or divide by a conversion factor. Course 1 9-3 Converting Customary Units
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Convert 3 quarts to cups. Additional Example 2: Converting Units of Measure by Using Proportions Convert quarts to cups . 1 x = 12 1 • x = 4 • 3 1 quart is 4 cups. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. Divide both sides by 1 to undo the multiplication. x = 12 3 quarts = 12 cups. Course 1 9-3 Converting Customary Units 3 qt = cups 1 qt 4 c 3 qt x =
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Convert 144 cups to gallons. Check It Out: Example 2 Convert cups to gallons . 16 x = 144 16 • x = 1 • 144 1 gallon is 16 cups. Write a proportion. Use a variable for the value you are trying to find. The cross products are equal. Divide both sides by 16 to undo multiplication. x = 9 144 cups = 9 gallons. Course 1 9-3 Converting Customary Units 144 cups = gallons . 1 gal 16 c x 144 c =
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Lesson Quiz 1. Convert 5 yards to inches. 2. Convert 16 tons to pounds. 3. Convert 11 quarts to cups. 4. A project requires 288 inches of tape. How many yards is this? 32,000 pounds 180 inches Insert Lesson Title Here 44 cups 8 yards Course 1 9-3 Converting Customary Units
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Lesson Quiz Convert. 1. A book is 24 cm long. 24 cm = ____ mm 2. The chain has a mass of 16 g. 16 g = _____ mg 3. The volume of the liquid was 12,000 mL. 12,000 mL = ____ L 4. Frank’s paper airplane glided 78.9 m. Sarah’s plane glided 85 m. How many more centimeters did Sarah’s plane glide? 16,000 240 Insert Lesson Title Here 12 610 cm Course 1 9-4 Converting Customary Units
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Additional Example 1: Finding the Perimeter of a Polygon Find the perimeter of the figure. 2.8 + 3.6 + 3.5 + 3 + 4.3 Add all the side lengths. The perimeter is 17.2 in. Course 1 9-7 Perimeter
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Lesson Quiz: Part I Find each perimeter. 1. 2. 3. What is the perimeter of a polygon with side lengths of 15 cm, 18 cm, 21 cm, 32 cm, and 26 cm? 9 cm 4 ft Insert Lesson Title Here 112 cm 5 6 __ Course 1 9-7 Perimeter
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Lesson Quiz: Part I Find each perimeter. 1. 2. 3. What is the perimeter of a polygon with side lengths of 15 cm, 18 cm, 21 cm, 32 cm, and 26 cm? 9 cm 4 ft Insert Lesson Title Here 112 cm 5 6 __ Course 1 9-7 Perimeter
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Warm Up Write impossible, unlikely, as likely as not, likely, or certain to describe each event. 1. A particular person’s birthday falls on the first of a month. 2. You roll an odd number on a fair number cube. 3. There is a 0.14 probability of picking the winning ticket. Write this as a fraction and as a percent. unlikely as likely as not Course 1 12-2 Experimental Probability , 14% 7 50 __
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Performing an experiment is one way to estimate the probability of an event. If an experiment is repeated many times, the experimental probability of an event is the ratio of the number of times the event occurs to the total number of times the experiment is performed. Course 1 12-2 Experimental Probability
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Check It Out: Example 2 For one month, Ms. Simons recorded the time at which her bus arrived. She organized her results in a frequency table. Time 4:31-4:40 4:41-4:50 4:51-5:00 Frequency 4 8 12 Course 1 12-2 Experimental Probability
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Check It Out: Example 2A Find the experimental probability that the bus will arrive before 4:51. = 4 + 8 24 _____ = 12 24 ___ = 1 2 __ Before 4:51 includes 4:31-4:40 and 4:41-4:50. P (before 4:51) number of times the event occurs total number of trials ___________________________ Course 1 12-2 Experimental Probability
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Lesson Quiz: Part II 2. Find the experimental probability that the spinner will land on blue. 3. Find the experimental probability that the spinner will land on red. 4. Based on the experiment, what is the probability that the spinner will land on red or blue? Insert Lesson Title Here Sandra spun the spinner above several times and recorded the results in the table. 2 9 __ 4 9 __ Course 1 12-2 Experimental Probability 2 3 __
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An experiment with equally likely outcomes is said to be fair . You can usually assume that experiments involving items such as coins and number cubes are fair. Course 1 12-6 Theoretical Probability
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When you combine all the ways that an event can NOT happen, you have the complement of the event. Course 1 12-4 Theoretical Probability
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Lesson Quiz Use the spinner shown for problems 1-3. 1. P (2) 2. P (odd number) 3. P (factor of 6) 4. Suppose there is a 2% chance of spinning the winning number at a carnival game. What is the probability of not winning? Insert Lesson Title Here 98% 2 7 __ 4 7 __ 4 7 __ Course 1 12-4 Theoretical Probability
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