UNIT 6 BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle The Circle 1
Questions How are perimeter and area related? How are the areas of polygons and circles related and applied? 2
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
Circle Terminology Circle Disc Radius Tangent Central Angle Sector Pi Chord Diameter Arc Circumference Area of a disc Perpendicular bisector Square Root 4
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
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
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
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
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
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
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
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
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
Discovering Circles Draw 3 circles passing through B & C. How many circles can pass through B & C? Infinite number 14
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
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
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 = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 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
First 5000 digits of PI 3. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455 34690 83026 42522 30825 33446 85035 26193 11881 71010 00313 78387 52886 58753 32083 81420 61717 76691 47303 59825 34904 28755 46873 11595 62863 88235 37875 93751 95778 18577 80532 17122 68066 13001 92787 66111 95909 21642 01989 18
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
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 = 18.84 cm d = C C = 100.48 cm d = C 20
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
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
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
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
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: 1. 45 = 7.85 cm 2. 60 3. 120 4. 10 C = 2 3.14 10 = 62.8 x_ = 45 62.8 360 x = 62.8 45/360 x = 2826/360 x = 7.85 cm 10cm 25
Central Angle is compared to 360 & length of the arc is compared to the circumference 2. 60 3. 120 4. 10 26
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) 25.12 m? 3) 5.02 m? 4) 4.2 m? C = 2 r C = 2 3.14 8 = 50.24 m 1.2 = x 50.24 360 x = 1.2 360/50.24 x = 8.6 27
Finding the measurement of a central angle 2) 25.12 m? 3) 5.02 m? 4) 4.2 m? 28
Area of a Square & Square Root Parts of the SQUARE ROOT to know: radical 64 = 8 square root radicand 29
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 16. 16 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
Square Root 31
Area of a Square & Square Root 32
To distinguish between a positive & a negative square root: x x Means positive square root. Means negative square root. 1. 25 4. 100 2. 49 5. 9/ 25 3. 1 6. 0.01 33
Square root questions 1. Square root of 300 17 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 = 0.3 34
Area of a DISC A = r 2 = 3.14 (9cm) 2 r = 9 cm = 3.14 (81cm 2 ) = 254.34 cm 2 35
Calculate the AREA of each disc 1. d = 13 cm 2. C = 35.6 cm C = d A = r 2 36
Calculate the AREA of each disc 3. Find the diameter of this circle. A = 452.16 cm 2 A = r 2 r 2 =A r 2 = 452.16 3.14 r 2 = 144 cm 2 r = 144 r = 12 cm d = 12 2 = 24 cm 37
Area of a Sector Measure of the Central Angle Area of the Sector sector 360 180 90 60 45 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. 18 10 n r 2 /20 r 2 /36 r 2 n/360 38
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
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
Find the area of each sector. 1. 2. 3. 75 100 r = 5cm r = 8cm r = 10cm 210 41
Valentine Flowers Basic Flower 42