Ch. 14 Area, Pythagorean Theorem, and Volume Section 14-1 Linear Measure

Transcription

1 Units of Length in the English System: Ch. 14 Area, Pythagorean Theorem, and Volume Section 14-1 Linear Measure 1 yard = 3 feet = 36 inches 1 foot = 1 /3 yard = 1 /5280 mile 1 foot = 12 inches 1 inch = 1 /12 foot = 1 /36 yard = 1 /63360 mile 1 mile = 1760 yards = 5280 feet 1 yard = 1 /1760 mile We can convert from one unit of measure to another using dimensional analysis. Example: Convert 64 inches into yards Example: Convert 0.4 miles into feet. Example: Convert feet into yards. Example: If a car is traveling at 65 miles per hour, what is its speed in feet per second? Example: Convert 5432 yards per minute into miles per hour 1

2 The Metric System Unit Symbol Relationship to base unit Kilometer km 1000 m Hectometer hm 100 m Dekameter dam 10 m Meter m Base unit Decimeter dm 0.1 m Centimeter cm 0.01 m Millimeter mm m Conversion Relationship Kilo 10 3 hecto 10 2 deka 10 1 base (unit) 10 0 deci 10-1 centi 10-2 milli 10-3 Example: Convert the following a) 1.7 km into meters b) 385 mm into meters c) 0.08 km into centimeters d) 15 mm into centimeters 2

3 Distance Properties: 1. The distance between any two points A and B is greater than or equal to 0. (AB 0) 2. The distance between any two points A and B is the same as the distance between points B and A. (AB = BA) 3. For any three points A, B, and C, the distance between A and B plus the distance between B and C is greater than or equal to the distance between A and C (AB + BC AC) If A, B, and C are collinear and B is in between A and C, then AB + BC = AC. A B C If A, B, and C are NOT collinear (and form a triangle), then AB + BC > AC. This is known as the Triangle Inequality. B A C Example: Two sides of a triangle are 31 cm and 85 cm long and the measure of the third side must be measured in centimeters. a) What is the longest the third side can be? b) What is the shortest the third side can be? 3

4 Perimeter and Circumference The perimeter of a simple closed curve is the length of the curve. Example: Find the perimeter of the following: a) A square with 5 mm sides. b) A rectangle with length 8 feet and width 3 feet. The perimeter of a circle is called circumference. In the late 18 th century, mathematicians proved that the ratio of circumference to diameter is pi. ie, C d = π So, C = or C = Example: Find the circumference of a circle with : a) A diameter of 12 inches b) A radius of 0.5 cm. Example: If the circumference of a circle is 20π ft, what is the radius of the circle? 4

5 Arc Length: The length of an arc on a circle depends on the radius and the central angle. Since a circle contains 360, then 1 is of a circle; 15 is of a circle; etc. θ The length of an arc whose central angle is Ө determines 360 of a circle. The central angle in a semi-circle is, therefore the arc length of a semi-circle is. The central angle in a quarter-circle is, therefore the arc length of a quarter-circle is. Therefore, an arc of Ө will have length. Example: Calculate the length of an arc with: a) Central angle 45 and a radius of 10 mm. b) Central angle 124 and a radius of 5 feet. 5

6 Section 14-2 Areas of Polygons and Circles Finding Areas on a Geoboard Examples: Find the area of the following figures Areas of Polygons: Area of a Rectangle: A= l w Area of a Parallelogram: A = Area of a Triangle: A = 6

7 Area of a Kite: A = Area of a Trapezoid: A = Area of a Regular Polygon: A = 7

8 Area of a Circle: A = Area of a Sector of a Circle: A = Examples: Find the area of each of the following: a) 2 ft 6 ft b) 10 mm 15 mm 8

9 c) 8 yd 4 yd 11 yd d) 1.5 cm 2.2 cm e) 3 ft 2 ft 9

10 f) 1.05 m g) 28.2 in h) 8 in 8 in 20 inches 10

11 Example: Complete the table and answer the questions that follow. Starting Dimensions 3 ft 2 ft Multiply length and width by 2 Length Width Perimeter Area Starting Dimensions 4 in 3 in Multiply length and width by 2 Starting Dimensions 10 yd 8 yd Multiply length and width by 2 a) When you double the length and width of the rectangle, does the perimeter double? b) When you double the length and width of the rectangle, does the area double? b) What happens to the area of a rectangle when you double the length and width? Example: If the ratio of the sides of two squares is 1 to 5, what is the ratio of their areas? 11

12 Converting Units of Area (metric): 1 m 100 cm 1000mm 1 m 100 cm 1000 mm 1 m km 1 m km Example: Complete the following conversions: a) 650 cm 2 into m 2 b) 650 cm 2 into mm 2 c) 35 km 2 into m 2 d) 90,000,000 mm 2 into km 2 12

13 Converting Units of Area (English): 1 ft 1 yd Unit of Area 1 ft in 2 or 1 yd 2 9 ft 2 or 1296 in 2 or 1 mi 2 3,097,600 yd 2 or 27,878,400 ft 2 1 mi (1760 yd) (5280 ft) Equivalent in other units yd2 or 27,878,400 mi2 1 3,097,600 mi2 Example: Complete the following conversions: a) 5000 ft 2 into yd 2 b) 150 yd 2 into ft 2 c) 10,000 yd 2 into mi 2 d) 25 mi 2 into ft 2 13

14 Land Measurement: 1 acre = 4840 yd 2 1 mi 2 = 640 acres Examples: Convert the following: a) 1 acre into ft 2 b) 11,520 acres into mi 2 c) 15 mi 2 into acres Section 14-3 The Pythagorean Theorem, Distance Formula, and Equation of a Circle The Pythagorean Theorem: If a right triangle has legs of lengths a and b and hypotenuse of length c, then a 2 + b 2 = c 2. 14

15 Examples: Find the missing side: a) x 8 5 b) 16 x 28 Example: Could the following sides be the sides of a right triangle? a) 2, 3, 13 b) 3, 4, 7 Example: Two cars depart from the same house at 5:00 pm. One drives south at 50 mph and the other drives east at 60 mph. At 8:00 pm, how far apart are the two cars? 15

16 What are Pythagorean Triples? Special Right Triangles In an isosceles right triangle with the length of each leg a, the hypotenuse has length. In a triangle, the length of the hypotenuse is twice the length of the leg opposite the 30 degree angle, and the leg length opposite the 60 degree angle is times the length of the shorter leg. The Distance Formula: The Distance between two points (, ) (, ) x y and x y is:

17 Example: Find the distance between the points ( 2,1) and ( 5,4) Circles are not functions, but we can still write an equation to represent the relationship between the x and y coordinates of every point on a circle. It is based on the distance formula. The Equation of a Circle with Center at the Origin: The Equation of a Circle with Center at (h, k): Examples: a) Write the equation of a circle centered at (6, 4) with a radius of 7. b) Determine the center and radius of the following circle: (x 3) 2 + (y + 4) 2 = 25 17

18 Section 14-4 Surface Area The surface area of a 3 dimensional figure is the sum of the areas of the lateral faces. Consider a cube and its net: The Surface Area of a Cube is: SA = Consider a rectangular prism and its net: The Surface Area of a Rectangular Prism is: SA = 18

19 Consider a pentagonal prism and its net: The Surface Area of a Pentagonal Prism is: SA = Consider a cylinder and its net: The Surface Area of a Cylinder is: SA = Consider the following right square pyramid and its net: The Surface Area of a Pyramid is: SA = 19

20 Consider a cone: The Surface Area of a Cone is: SA = Consider a sphere: The Surface Area of a Sphere is: SA = Examples: Find the surface area of the following figures: a) 4 in 20 in 2 in 20

21 b) 3 cm 5 cm c) 0.75 ft 1.5 ft d) The following snow-cone 2 in 5 in 21

22 e) 6 cm 4 cm 4 cm e) A sphere with diameter of 20 in. Example: How much material is needed to make the following tent? 3 ft 6 ft 5 ft 22

23 Example: How does the surface area of a box change if each dimension is doubled? Examples: How does the surface area of a right circular cone change if you triple the radius and triple the slant height? Section 14-5 Volume, Mass, and Temperature Converting Measures of Volume: (most common are cm 3 and m 3 ) Each metric unit of length is 10 times as great as the next smaller unit. Each metric unit of area is 100 times as great as the next smaller unit. Each metric unit of volume is 1,000 times as great as the next smaller unit. 1 m 3 = 1,000,000 cm 3 (to go from m 3 to cm 3, move decimal 6 places right) 1 cm 3 = m 3 (to go from cm 3 to m 3, move decimal 6 places left) 1 in 3 = ft3 1 ft 3 = 1 27 yd3 1 yd 3 = 27 ft 3 23

24 Examples: Convert the following: a) 9 m 3 = cm 3 b) 13,400 cm 3 = m 3 c) 45 yd 3 = ft 3 d) 4320 in 3 = ft 3 = yd 3 We generally use Liters (ml, L, kl, etc. for liquid measurement) 1 L = 1000 cm 3 1 cm 3 = 1 ml a) 27 L = ml b) 3 ml = cm 3 c) 5 m 3 = cm 3 = L 24

25 Volume of a right Rectangular Prism: V = h l w Volumes of all right Prisms and Cylinders: V = Bh (B = area of the base) Examples: Find the volume of the following: a) 7 cm b) 2 in 4 in c) 8 yd 5 yd 2 yd 25

26 Volume of a right Pyramid: V = Volume of a right Cone: V = Examples: Find the volume of the following: a) 1.5 in 4 in 6.5 in b) 12 cm 6 cm Volume of a Sphere: V = Example: Find the volume of a sphere with diameter 8 meters. 26

27 What happens to volume if you change the dimensions (similar figures)? Example: Suppose a cube measures 4 inches along each side. What happens to the volume if you double the dimensions? What if you triple them? Mass: The fundamental unit for mass is the gram. 1 kilogram 1 gram 1 milligram g g Examples: Convert the following a) 64 g = kg b) 7524 kg = g c) 580 g = mg Temperature (Fahrenheit vs. Celsius) ( ) C = F 32 = F or F = C Examples: Convert the following: a) 80 F = C b) 10 C = F c) 212 F = C d) 0 C = F 27

28 Attributes and Units Measurement is a three-step process: choose an attribute to measure, choose an appropriate unit, determine how many of these units are necessary to find the length, cover, or fill the object. In this activity you will focus on the attribute and the unit. Next to each description are two blanks. In the first blank, label the attribute that is being measured. Attribute choices: L (length), A (area), SA (surface area), V (volume) In the second blank, choose the best unit. Unit Choices: mm, m, km (length) in 2, ft 2, acre (area or surface area) ml, L, kl (volume) Problem Attribute Unit Description 1 L Km The distance from New York City to Chicago 2 The amount of wrapping paper needed to wrap a CD 3 The height of a 5 story building 4 The width of a cockroach 5 The amount of tea in a pitcher 6 The amount of space covered by a bathroom floor 7 The amount of gas in a car s full gas tank 8 The amount of grass in Central Park 9 The amount of fabric needed to cover a couch cushion 10 The width of a two car garage 11 The wingspan of a hummingbird 12 The amount of wall space covered by a light switch 13 The amount of liquid held by a baby bottle 14 The size of a ceiling to be painted 15 The amount of paint needed to paint a ceiling 16 The amount of water in a hot tub 17 The thickness of an iphone 18 The size of a living room rug 19 The size of a label on a soup can 20 The amount of aluminum foil needed to cover a baked potato 28