UNIT 6. BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle. The Circle

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1 UNIT 6 BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle The Circle 1

2 Questions How are perimeter and area related? How are the areas of polygons and circles related and applied? 2

3 Main Ideas 1. The elements of a circle; 2. The relation between a circle and its diameter; 3. The concepts of perimeter and area 4. The characteristics and properties of a circle. Objective: To solve problems involving circles! 3

4 Circle Terminology Circle Disc Radius Tangent Central Angle Sector Pi Chord Diameter Arc Circumference Area of a disc Perpendicular bisector Square Root 4

5 Learning about circles What is the definition of a CIRCLE? Circle is a closed line with all points equidistant from an interior point known as the center. What is the definition of a DISC? Disc is the region of a plane comprising the circle and the interior. 5

6 Learning about circles Roll the disc along your ruler. Your ruler is tangent to the circle. How would you define tangent to the circle? A tangent is a line that contacts a circle at one point only. 6

7 Learning about circles Fold the disc so that both halves are perfectly superimposed. What is the fold called? The diameter What do you notice about the fold and the centre of the circle? The diameter passes through the center. What is the relation between the diameter and the radius of a circle? The diameter is equal to twice the radius or the radius is half the diameter. 7

8 Learning about circles Fold the disc in four, then unfold it. 1. What do you notice about the two folds that we just created? They are diameters, they intersect at the center and are perpendicular. 2. What are the properties of the diameter of a circle? All are congruent It passes through center Each diameter measures twice the radius of the circle It is the longest segment joining 2 points on the circle The diameter is an axis of symmetry. 8

9 Learning about circles Fold the disc so that the line forming the circle touches the centre, then unfold it. What is the fold called? A chord A chord is a line segment joining 2 points on a circle What is the part of the circle defined by the fold called? An arc of a circle An arc is a part of the circle that is created by a chord or 2 radii 9

10 Learning about circles Fold the disc so that half of the previous chord coincides with the other half. Then unfold the disc. What is the fold in relation to the chord called? Perpendicular Bisector of chord Fold Does the fold pass through the centre of the circle? Yes 10

11 Learning about circles Fold the disc again so that the line forming the circle touches the centre. Fold the disc twice more in the same fashion. What figure is formed? Equilateral triangle Describe the characteristics of the figure. 3 sides 3 angles The 3 vertices are located on the circle at equal distances from one another. 11

12 Learning about circles How do you construct the perpendicular bisector of a line segment with a compass? 1. Open compass to more than half of line length, place it on Point A, draw a small arc on each side of line segment. 2. Repeat the same from Point B, arcs should intersect creating new points. 3. Draw a line segment from new point to new point. 12

13 Discovering Circles Draw a circle passing through Point A. Draw 3 more circles passing through A. How many different circles can pass through A? Infinite number 13

14 Discovering Circles Draw 3 circles passing through B & C. How many circles can pass through B & C? Infinite number 14

15 Discovering Circles How can we find the centre of the circle that passes through points A, B & C? A B 1. Draw a line from A to B, then bisect it. 2. Draw a line from B to C, then bisect it. C 3. Extend the bisectors so they intersect. Where they meet is the center of the circle that all 3 points are part of. 4. Draw the circle. 15

16 Important to remember that: 1. Three non-collinear points determine one and only one circle. *Non-collinear means not on the same line!* 2. The diameters (or radii) of a circle are congruent ( ). 3. In a circle, the length of the radius is half the length of the diameter. r = d or d = 2r 2 4. The axes of symmetry of a circle pass through the center. 5. The longest chord in a circle is the diameter. 6. The perpendicular bisectors of the chords of a circle intersect at the center. 7. The perimeter of a circle is called the CIRCUMFERENCE. 16

17 Perimeter or Circumference of a Circle The relation between circumference and the diameter of a circle can be expressed by: C = which is equivalent to C = d d or C = 2 r The number is a non-repeating, non-terminating decimal = Usually, is assigned the approximate value of 3.14, and so the circumference of a circle is approximate, whether it is calculated from the diameter or the radius. 17

18 First 5000 digits of PI

19 Perimeter or Circumference of a Circle C = d or C = 2 r Find the CIRCUMFERENCE of each circle. d = 13 cm r = 12.5 cm C = d C = 2 r 19

20 Thinking about circles If C = d or C = 2 r, how can the following be expressed: 1) d = C 2) r = C 2 Find the DIAMETER of each circle. C = cm d = C C = cm d = C 20

21 Pi Facts 1995, Hiroyuki Goto from Japan recited fortytwo thousand, one hundred, ninety-five digits without error. Now there's a geek! Pi is the ratio of circumference and diameter Pi is an irrational number As if being this important circumferencediameter ratio isn't enough, pi can also be used to find the area of a circle! 21

22 Measurement of an ARC The measure of an arc is also expressed in units of length. We use the measure of the central angle & the circumference of the circle to determine its measure. C = d 80 or C = 2 r 22

23 A Proportional Situation is found Arc length = Central Angle 30 r = 8 m Circumference 360 So, if we let x represent arc length X = C A C 360 C = d 23

24 A Proportional Situation is found 100 Find the arc length of a 100 central angle in a circle with a radius of 4 cm? Arc length Circumference = Central Angle 360 r = 4 cm X = C A C 360 C = d 24

25 Central Angle is compared to 360 & length of the arc is compared to the circumference In a circle with a radius of 10cm, find the measure of the arc intercepted by a central angle of: = 7.85 cm C = = 62.8 x_ = x = /360 x = 2826/360 x = 7.85 cm 10cm 25

26 Central Angle is compared to 360 & length of the arc is compared to the circumference

27 1) 1.2 m? Finding the measurement of a central angle In a circle with a radius of 8 m, what is the measure of a central angle created by an arc of: = 8.6 Arc length Circumference = Central Angle 360 2) m? 3) 5.02 m? 4) 4.2 m? C = 2 r C = = m 1.2 = x x = /50.24 x =

28 Finding the measurement of a central angle 2) m? 3) 5.02 m? 4) 4.2 m? 28

29 Area of a Square & Square Root Parts of the SQUARE ROOT to know: radical 64 = 8 square root radicand 29

30 Area of a Square & Square Root Squaring is an inverse operation. The operation is called taking the square root. The symbol used for this operation is, this is a radical sign. Example: is read the square root of The square root of a number is an operation that finds a number that when multiplied by itself gives the radicand. Thus, 4 4 = 16 30

31 Square Root 31

32 Area of a Square & Square Root 32

33 To distinguish between a positive & a negative square root: x x Means positive square root. Means negative square root /

34 Square root questions 1. Square root of Yes or No??? How can we check this? 2. What does your calculator display when you try to calculate the square root of -4? Explain this result. 3. Can the square root of a positive fraction be found? Yes or No Give an example. 9/16 = 3/4 4. Can the square roots of positive decimal values be found? 0.09 Give an example =

35 Area of a DISC A = r 2 = 3.14 (9cm) 2 r = 9 cm = 3.14 (81cm 2 ) = cm 2 35

36 Calculate the AREA of each disc 1. d = 13 cm 2. C = 35.6 cm C = d A = r 2 36

37 Calculate the AREA of each disc 3. Find the diameter of this circle. A = cm 2 A = r 2 r 2 =A r 2 = r 2 = 144 cm 2 r = 144 r = 12 cm d = 12 2 = 24 cm 37

38 Area of a Sector Measure of the Central Angle Area of the Sector sector r 2 r 2 /2 r 2 /4 r 2 /6 r 2 /8 Is this a proportional situation? Give an expression to represent the area of a sector with a central angle of n n r 2 /20 r 2 /36 r 2 n/360 38

39 Area of a Sector Proportion We use a proportion to calculate the area of a sector. sector n disc Sector Disc Angle Area n = x 360 πr 2 39

area that measures the area of the disc. 2.

area that measures the area of the disc. 3.

40 Fill in the blank 1. A central angle that measures 180 has an area that measures the area of the disc. 2. A central angle that measures 120 has an area that measures the area of the disc. 3. A central angle that measures 90 has an area that measures the area of the disc. 4. A central angle that measures 60 has an area that measures the area of the disc. 5. A central angle that measures n has an area that measures the area of the disc. 40

41 Find the area of each sector r = 5cm r = 8cm r = 10cm

42 Valentine Flowers Basic Flower 42

Circles. Parts of a Circle: Vocabulary. Arc : Part of a circle defined by a chord or two radii. It is a part of the whole circumference.

Circles. Parts of a Circle: Vocabulary. Arc : Part of a circle defined by a chord or two radii. It is a part of the whole circumference. Page 1 Circles Parts of a Circle: Vocabulary Arc : Part of a circle defined by a chord or two radii. It is a part of the whole circumference. Area of a disc : The measure of the surface of a disc. Think