Section 4.1 Investigating Circles

Transcription

1 Section 4.1 Investigating Circles A circle is formed when all the points in a plane that are the same distance away from a center point. The distance from the center of the circle to any point on the edge is called the radius. The distance across the circle through the center is called the diameter. Note: A circle is named by the point in the center. A radius is a named by its endpoints. A diameter is named by its endpoints. circle: F radius: FO, FR or FG diameter: RG Example 1: Write the name of each circle, radius and diameter. a) b) circle: radius: diameter: circle: radius: diameter: L. Brenton Page 1

2 Since the distance from the center to the edge of a circle is the radius and the diameter is the distance from one edge of the circle to the other passing through the center, we can easily see that two radii make up one diameter. Therefore, given the radius (r) of a circle, we can determine the diameter (d) and vice versa. For example, a) if r = 5 m, then d = b) if d = 14.2 cm, then r = Example 2: Write the radius and diameter for each. a) b) c) radius: diameter: radius: diameter: radius: diameter: d) e) f) radius: diameter: radius: diameter: radius: diameter: L. Brenton Page 2

3 Example 3: The merry-go-round on the school playground measures 3 m from the center to the edge. What is the diameter of the merry-go-round? Example 4: A carpenter is making a table. For the top of the table, she wants to cut a circle from a square piece of board that is 2 m in length. What is the diameter of the largest circle she can cut? What is the radius? Example 5: Mark the center in each circle below and measure the radius and diameter. a) b) c) L. Brenton Page 3

4 Section 4.2 Circumference of a Circle The distance around a figure is called the perimeter. The perimeter of a circle has a special name the circumference. To see the relationship between the circumference of a circle and its radius and diameter, complete the following investigation. Investigation: Using a string and ruler, measure the circumference, radius and diameter of the six circular objects as indicated by your teacher and record the information in the table below. Use a calculator to complete the last column. Object Circumference (cm) Radius (cm) Diameter (cm) Circumference Diameter L. Brenton Page 4

5 Questions: A) Do you notice a pattern between the diameter and the circumference of each circle? Explain what you observe. B) Do you notice a pattern between the radius and the circumference of each circle? Explain what you observe. C) What do you notice about the circumference diameter for each circle? D) Does the pattern in circumference diameter depend on the size of the circle? L. Brenton Page 5

6 For each object, notice that each time the circumference of the object is divided by the diameter, you should have gotten a number that was close to 3. That means the circumference of the object is about 3 times its diameter. Therefore, we can an estimate of the circumference of a circle by multiplying the diameter by 3. Circumference 3 d Example 1: Given each diameter, estimate the circumference of the circles below. a) b) c) Example 2: Given each radius, estimate the circumference of the circles below. a) b) c) L. Brenton Page 6

7 If given the circumference of a circle, we can reverse the equation to find the diameter and the radius. d C 3 Example 3: Given each circumference, estimate the diameter and radius of the circles below. a) A circular ground has a circumference of 30ft. b) The circumference of a bike wheel is 27 m. c) A circle has a circumference of 33.9 ft. d) A string of length 18 mm is bent to form a circle. L. Brenton Page 7

8 When the exact circumference of a circle is divided by its exact diameter, the answer will always be the same for a circle of any size. This number is known as PI. We represent this number with the symbol π. We use a symbol to represent PI because it is a number that goes on forever and never repeats itself in a pattern. Numbers such as this are called irrational numbers. Here are the first 100 digits of pi. To find the circumference and diameter of a circle more accurately, we should use π instead of 3. However, since pi never ends, we will round pi to 3.14 for use in calculations. Therefore, π 3.14 and the correct formulae are C = πd and d = C π L. Brenton Page 8

9 Example 1: Find the circumference of each circle below to one decimal place. a) b) c) d) e) f) Example 2: a) A circle has diameter 10.5 cm. Find the circumference to the nearest millimetre. L. Brenton Page 9

10 b) A circle has radius 4.3 mm. Find the circumference to the nearest millimetre. c) A circle has circumference 12.6 m. Find the diameter to the nearest centimetre. Example 3: A circular tablecloth has diameter 1 m. The designer wants to put a fringe around the edge of the cloth. How much fringe should he buy, if fringe is sold by the tenth of a meter? Explain. L. Brenton Page 10

11 Section 4.3 Area of a Parallelogram What is Area? Area can be defined as a measure of the space inside a region, or how many square units it takes to cover a region. Since area is 2-dimensional, it is measured in square units These may include km 2, hm 2, dam 2, m 2, dm 2 cm 2, mm 2 Recall: Area of a Rectangle A rectangle is a closed, four-sided figure made up of four right angles and two sets of equal, parallel sides. To find the area of a rectangle we multiply its length by its width, A = l w. For example, to find the area of the rectangle below, we multiply the length and width. 10 cm Area = l w 6 cm = = Example: Find the area of the following rectangles. 8 cm a) b) 5 cm 6.2 m 9.5 m L. Brenton Page 11

12 What is a Parallelogram? A parallelogram is a four side figure with both pairs of opposite sides parallel. Parallel lines are always an equal distance apart and therefore never intersect each other. NOTE: An equal number of arrow heads indicate which lines are parallel. Examples of Parallelograms Not Examples of Parallelograms Parts of a Parallelogram The height is always where a 90 o angle forms with the base. L. Brenton Page 12

13 Area of a Parallelogram If we were to cut any parallelogram along its height and reconnect it by joining one set of parallel sides we would create a rectangle. h h b b The length and width of the rectangle made becomes the height and base of the rectangle. Therefore, similar to finding the area of a rectangle, we multiply the base and height to find the area of a parallelogram. For example, to find the area of the parallelogram below, we multiply the base and height. Area of a Parallelogram = b h Let s try the following example! Area = b h = = L. Brenton Page 13

14 Example 1: Find the area of each parallelogram below a) b) c) d) e) f) L. Brenton Page 14

15 Example 2: Find the base or height of each parallelogram. a) b) c) d) e) f) L. Brenton Page 15

16 Section 4.4 Area of a Triangle What is a triangle? A triangle is closed figure with 3 sides and 3 enclosed angles. Parts of a Triangle Base (b): any side of a triangle Height (h): the distance from the base to the opposite vertex. NOTE: Each point created by an angle on a triangle is called a vertex. More than one are called vertices. Important!!! The base and height are always the two measures that form a 90 angle. L. Brenton Page 16

17 Determining the Formula for Area of a Triangle We can make two congruent triangles from any parallelogram just by cutting along a diagonal. Notice that the area of each triangle is half of the area of the parallelogram we started with. Therefore, if the area of a parallelogram is b h then the area of a triangle is half of that. Area of a Triangle = 1 (b h) 2 Let s try the following example! Area = 1 (b h) 2 = = L. Brenton Page 17

18 Example 1: Find the area of each triangle. a) b) c) d) e) f) L. Brenton Page 18

19 Example 2: Given the area of each triangle, find the base or height. a) b) c) d) e) f) L. Brenton Page 19

20 Example 3: Daniel just bought a used sailboat with two sails that need replacing. How much fabric will Daniel need if he replaces sail A? Example 4: Daisy wants new flooring and carpeting for her rectangular apartment. A floor plan of her apartment is shown below. If kitchen flooring costs $12.95 per square meter, how much will it cost Daisy to put new flooring on her kitchen? L. Brenton Page 20

21 Section 4.5 Area of a Circle Remember, to find the circumference of a circle, we use C = πd We can use this knowledge to develop a formula for finding the area of a circle. Divide a circle into equal sectors. Ex. If we cut out each sector and arrange in a shape resembling a parallelogram we get something similar to the picture shown below Note: - the height of the parallelogram is the same as the radius of the circle, r - the base of the parallelogram is the same as half the circle s circumference, πd When we replace the height of the parallelogram with r and the base of the parallelogram with πr, we get 2 or πr Area of a Circle = πr 2 L. Brenton Page 21

22 Example 1: Find the area of each circle. a) b) c) d) e) f) L. Brenton Page 22

23 Example 2: A carpenter is making a circular tabletop with radius 0.5 m. What is the area of the tabletop to the nearest tenth of a meter? Example 3: The diameter of a button on an mp3 player is 0.78 cm. a) What is the radius of the button? b) What is the circumference of the button? c) What is the area of the button? Example 4: The circular vent on a furnace has diameter 19.4 cm. What is the area of the vent? L. Brenton Page 23

24 Now that we know the formulae for the area of various shapes, we can consider more complex problems. Consider the diagram to the left. How would we find the area of the shaded region? The complex figure can be broken into two simple polygons. Therefore, we can find the area of both shapes separately and then add or subtract as the problem indicates. For this example, we would subtract. Therefore, the area of the shaded region would be equal to the area of the big rectangle minus the area of the small rectangle. Let s complete the problem! Area of the Area of the Area of the Big Rectangle Small Rectangle Shaded Region L. Brenton Page 24

25 Example 5: Find the area of the shaded region for each composite figure. a) b) c) d) L. Brenton Page 25

26 Section 4.6 Interpreting Circle Graphs A circle graph compares amounts by using area sectors of a circle. Data is shown as a fraction (or percent) of the area. The circle represents one whole or 100% of a set of data. Each sector or part of a circle graph represents a percent of the whole. The title, legend and labels are crucial to interpreting circle graphs. The graphs can be labeled with the actual data and/or percents. Each category sector must be labelled. Each piece of the circle is called a SECTOR. The box showing what category each sector represents is called the LEGEND In order to understand what a circle graph is displaying we need to be able to analyze the data being presented. L. Brenton Page 26

27 Example 1: The annual energy bill for a typical single family home is shown in the circle graph. If the average energy bill is $2200 per year, how much is spent on each of heating, electronics and water heating? Heating: If heating is 29% of the energy bill, how much of the $2200 is spent on heating? This means, 29% of Electronics: Water Heating: L. Brenton Page 27

28 Example 2: This circle graph shows how much time is spent in one day watching different types of TV programs. a) Which type of program is watched for the greatest amount of time? b) Which two types of programs are watched for approximately the same amount of time? c) Estimate the fraction of time spent watching sitcoms. d) Suppose TV is watched for 1000 days. Estimate how much time is spent watching sitcoms. L. Brenton Page 28

29 A circle s central angles add up to 360. Section 4.7 Drawing Circle Graphs When a circle is divided into 100 equal parts each sector is 1%. 1% is equal to 3.6 To draw circle graphs you need to be able to use a protractor to draw angles of a certain measure. Use a protractor to draw the following angles. a) 40 b) 102 c) 228 Example 1: Using a protractor, divide the circle below into 5 sectors with the following angle measures. 35, 80, 60, 135, 50 L. Brenton Page 29

30 Example: For each circle below, find the missing angle measures. a) b) The circle graph below shows the percent of students who chose different options for lunch. Determine the measure of each central angle in degrees. Since there are 360 in a circle we need to find each percent of % of % of % of % of 360 L. Brenton Page 30

31 To construct a circle graph from a data table there a three basic steps: convert the data to fractions convert the fraction to decimals find the percent as an angle Example: A survey was conducted among 27 Grade 9 students at Learn More High School to find out students favorite kind of music. The results are shown below. Construct a circle graph for this data. Type of Music Frequency (27 total) As a Fraction As a Decimal % as an angle (% of 360) Classical 2 2/ = 7% 0.07 X 360 = Rap/Hip Hop 7 Rock 11 Country 4 Jazz/Blues 3 LEGEND L. Brenton Page 31

32 Conduct a survey to see what flavor of ice cream your classmates prefer. Record the data in the table below. Flavor Tally Frequency ( total) As a Fraction As a % % as an angle (% of 360) Strawberry Chocolate Neopolitan Vanilla Pineapple Total TITLE: LEGEND L. Brenton Page 32