11.2 Area of Circles and Sectors Geometry

Geometry 11.2 Areas of Circles and Sectors Goals Find the area of a circle. Find the area of a circle sector. Solve problems using areas. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors The Area of a Circle Finding the area of a circle is harder than you might think. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors 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 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors 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. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Method 3: Use Triangles February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Method 3: Use Triangles Separate the top half from the bottom half. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Method 3: Use Triangles Peel the wedges apart and line them up. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Method 3: Use Triangles Slip them together. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Method 3: Use Triangles (Allowing for poor artistic rendering….) This shape looks like a ____________. parallelogram The height is the same as the ______ of the circle. radius The length of the base is the same as the length of a _________ of the circle. semicircle r r February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Method 3: Use Triangles r2 The area of the parallelogram is A = ____. r r February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Area of a Circle r February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Example 1 Find the area of the circle. Solution: A = (72) = 49 in2 or A  153.94 in2 7 in February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Example 2 The area of a circle is 400 ft2. Find the radius of the circle. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors 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? February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Large Diameter = 16 Radius = ? Medium Diameter = 10 Radius = ? 8 5 5 8 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Large Radius = 8 Area = ? Medium Radius = 5 Area = ? 64 25 5 8 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Large Area = 64 Medium Area = 25 5 8 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

50 8 5 5 Large Area = 64 Medium Area = 25 So two medium pizzas have an area of 50 8 5 5 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Solution: Buying one large has more area than two mediums. 64 8 25 25 201 in2 5 5 157 in2 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Circle Sectors Circle Sector A circle sector is the region bounded by two radii and the intercepted arc of a circle. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Area of Circle Sectors The area of a circle sector is proportional to the measure of the central angle. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Area of Circle Sectors The area of a circle sector is proportional to the measure of the central angle. x r February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Example 4 Find the area of the sector. 50 10 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Your Turn Find the area of sector ABC. A 125 C B 4 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Example 5 The area of a sector of a circle is 32. What is the measure of the central angle if the radius of the circle is 6? February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Problem Solving Read the problem. What do you need to know to solve to? What information is given that you can use? Identify the steps needed. Solve it and answer the question. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Problem To water a square yard, a man installs four sprinkler heads as shown. The sprinkler heads spray water in a circular pattern with a radius of 4 feet. After a while, he notices some the grass is dying. What is the area of the yard that doesn’t get any water? 4 ft February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Problem Solution Area of square A = 162 = 256 ft2 Area of one circle A = (42) = 16 ft2 Area of four circles A = 4(16) = 64 A  201 ft2 4 ft 16 ft ? 8 ft ? 16 ft ? Dying Area 256 – 201 = 55 ft2 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Your Turn Find the area of the (ring) – the shaded region. 5 in 2 in February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Solution Find the area of the (ring). Area of Outer Circle Radius = 7 in. Area = 49 Area of Inner Circle Radius = 5 Area = 25 25 5 in 2 in 7 49 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Solution Outer Circle Area = 49 Inner Circle Area = 25 Area Outer – Inner = 49 - 25 = 24  75.4 in2 25 5 in 2 in 49 February 17, 2019 Geometry 11.2 Areas of Circles and Sectors

Geometry 11.2 Areas of Circles and Sectors Summary Area of a circle = r2 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. February 17, 2019 Geometry 11.2 Areas of Circles and Sectors