Warmup. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 1

Transcription

1 Warmup April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 1

2 Geometry 11.2 Area of Circles and Sectors

3 11.2 Essential Question How can you find the area of a sector of a circle? April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 3

4 Goals Use the formula for the area of a circle Use the formula for population density Find the areas of sectors Use areas of sectors April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 4

5 The Area of a Circle Finding the area of a circle is harder than you might think. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 5

6 Method 1: Archimedes Start with a circle. Inscribe a hexagon. Circumscribe a hexagon. Find the area of each hexagon. The area of the circle is average of the area of the two hexagons. Method of Exhaustion April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 6

7 Method 1 The more sides on the polygon, the more accurate the area will be. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 7

8 Method 2: Calculus A x Although this looks scary now, many of you will take calculus and discover that this is actually pretty simple to understand. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 8

9 Method 3: Use Triangles Start with a circle: Divide it into a number of wedges. As the number of wedges is increased, the shape of each gets closer and closer to an isosceles triangle. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 9

10 Method 3: Use Triangles April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 10

11 Method 3: Use Triangles Separate the top half from the bottom half. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 11

12 Method 3: Use Triangles Peel the wedges apart and line them up. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 12

13 Method 3: Use Triangles Slip them together. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 13

14 Method 3: Use Triangles (Allowing for poor artistic rendering.) This shape looks like a. parallelogram The height is the same as the radius of the circle. The length of the base is the same as the length of a semicircle of the circle. r r April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 14

15 Method 3: Use Triangles The area of the parallelogram is A =. r 2 r r April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 15

16 Area of a Circle A r 2 r April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 16

17 Example 1 Find the area of the circle. 7 in Solution: A = (7 2 ) = 49 in 2 or A in 2 April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 17

18 Example 2 The area of a circle is 400 ft 2. Find the radius of the circle. A r r r r 2 r r April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 18

19 Example 3 At Moldy Cheese Pizza they sell two sizes of pizza: medium and large. The medium has a diameter of 10 inches and the large has a diameter of 16 inches. On Fridays they have a special: you get one large pizza, or two medium pizzas for the same price. Which is the better deal? April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 19

20 Large Diameter = 16 8 Radius =? Medium Diameter = 10 5 Radius =? 8 5 April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 20

21 Large Radius = 8 Area =? 64 Medium Radius = 5 Area =? April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 21

22 Large Area = 64 Medium Area = April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 22

23 Large Area = 64 8 Medium Area = 25 So two medium pizzas have an area of April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 23

24 Solution: Buying one large has more area than two mediums Of course, with one pizza, everyone gets some anchovy in in April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 24

25 Circle Sectors Circle Sector A circle sector is the region bounded by two radii and the intercepted arc of a circle. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 25

26 Area of Circle Sectors The area of a circle sector is proportional to the measure of the central angle. x r A r 2 x 360 A = x Solve for A. 360 πr2 April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 26

27 Example 5 Find the area of the sector. A = x 360 πr2 A = π102 A = π 1 A = π 1 A = 125π 9 A Exact Answer! Approx. Answer! April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 27

28 Your Turn 2 A = x 360 πr2 A = π42 A = π 1 A = π 1 Find the area of sector ABC. A 125 B 4 A = 50π C April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 28

29 Example 6 The area of the sector of a circle is 32. What is the measure of the central angle if the radius of the circle is 6? A = x 360 πr2 32 = x 360 π62 32 = x π 1 32 = xπ π 10 π x = 320 π April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 29

30 Your Turn 3 Find the area of the annulus (pink ring). 5 in 2 in April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 30

31 Solution Find the area of the annulus (ring). Area of Outer Circle 25 5 in 2 in 7 49 Radius = 7 in. Area = 49 Area of Inner Circle Radius = 5 Area = 25 April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 31

32 Solution Outer Circle Area = 49 Inner Circle Area = 25 Annulus Area 25 5 in 2 in Outer Inner = = in 2 49 April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 32

33 Summary Area of a circle = r 2 Area of a sector is proportional to the measure of the central angle (or the measure of the intercepted arc). Leave answers in terms of pi when possible. April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 33

34 Homework April 28, 2017 Geometry 11.2 Areas of Circles and Sectors 34