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I like pie! Do you like pie? What is the shape of pie? Circles Using that special number! Circles A bicycle odometer uses a magnet attached to the wheel and a sensor attached to the bicycle frame. Each time the magnet passes the sensor, the odometer registers the distance traveled. This distance is the circumference of the wheel. Here is an example of the odometer on my bicycle. The pictures show the odometer reading and the sensor on the wheel. Circle: The set of all points in a plane that are equal distance from the center. (a circle is named by it’s center) Radius: a line segment with one endpoint at the center of the circle and the other endpoint on the circle. Radii is pleural of radius. Diameter: a line segment with endpoints on the circle and it passes through the center of the circle. Chord: a line segment with its endpoints on a circle. Do you like pie? Describe the shape of a pie. This will help you remember an important number called pi. This special number will be used to solve problems involving circles. Pi: (π) the ratio of the circumference of a circle to it’s diameter. Pi is an irrational number that is approximated by the rational numbers π ≈ 3.14 or 22/7. π ≈ 3.14159265358979323846…, although Pi has been computed to more than a trillion places, for circumferences and area using 3.14 is usually enough. Pi can be found to over a trillion places using computers today. Below is a fraction of what the number looks like when finding Pi. Circumference: the distance around a circle. Like perimeter, the circumference is the distance around the outside of a figure (circle). Unlike perimeter, in a circle there are no straight segments to measure, so a special formula is needed. C = πd when you know the diameter C = 2πr when you know the radius Area of a circle: the number of square units needed to cover the surface of a figure. Again a special formula is needed because there are no straight segments to measure. A = πr2 Find the area of the circle: when working with circles, be sure you are using the radius, i.e., A = πr². In this diagram, 10 is the diameter. The radius is half of the diameter. Example: Here is an example of a problem you will be asked to solve using circle formulas. In this figure you are asked to find the area of the shaded portion. There are many applications used everyday using formulas for circles. Take the example below. A Ferris wheel has a diameter of 56 feet and makes 15 revolutions per ride. How far would someone travel during a ride? C = πd C = πd C = 175.9291886… 175.9291886 · 15 = 2638.937829 feet Solving word problems Many car tire manufactures guarantee their tires for 50,000 miles. If the average tire has a 2 ft diameter, how many revolutions does the manufacturer guarantee? 1 revolution C = πd C = 3.14 · 2 C = 6.28ft Guaranteed mileage Feet/mile 5280/6.28 · 50,000 ≈ 42,038,216.56 revolutions Solving word problems Graph a circle with center (3, 1) that passes through (3, -1). Find the area and circumference, both in terms of π and to the nearest tenth. Use 3.14 for π A = πr2 C = πd A = π · 22 C = π · 4 A = 4π C = 4π A = 4 · 3.14 C = 4 · 3.14 A=12.56 units2 C=12.56units Area & Circumference of Circles Remember to use that special number called pi 1) Find the area of the circle. nearest tenth) a) b) c) d) 314.2 units 78.5 units 212 units 31.4 units (round to the 2) Find the circumference and area of a circle with a diameter of 15 cm. a) b) c) d) A = 126.9 cm2 C = 40.4 cm A = 176.6cm2 C = 47.1 cm A = 452.2 cm2 C = 144 cm A = 153.9 cm2 C = 44.0 cm 3) Find the circumference and area the circle. a) b) c) d) A = 6.2m2, C = 1.9m A = 7.1m2, C = 2.3m A = 4.5m2, C = 7.5m A = 50.2m2, C = 25.1m 4) Find the number of square inches in the area of the shaded region of this square. a) b) c) d) 95.0 sq. in. 285.1 sq. in. 25.9 sq. in 78.4 sq. in 5) Find the area of this figure. (round to the nearest square inch) a) b) c) d) 356.5 sq. in. 457.1 sq. in. 658.1 sq. in. 1060.2 sq. in. 6) Find the area of this figure. (round to the nearest square cm) a) b) c) d) 56.5 sq. cm 140.5 sq. cm 310.2 sq. cm 478.2 sq. cm 7) Butch, the dog, is leashed to the corner of the house when he is outdoors alone. The leash is 20 feet long. Find the amount of ground area available for Butch to run around in. (assume the corner of the house to be a right angle). a) b) c) d) 251.3 sq. ft. 314.2 sq. ft. 628.3 sq. ft. 942.5 sq. ft. 8) Find the area of the shaded portion of the circle. (round to the nearest tenth of a meter) a) b) c) d) 49.0 m2 153.9 m2 197.3 m2 297.7 m2 9) Find the radius of a circle with an area of 169 πin2. a) b) c) d) 26 in. 13 in. 9 in. 4 in. 10) Graph a circle with center (3, -1) that passes through (0, -1). Find the area and circumference, to the nearest tenth. a) b) c) d) A = 176.6 units2, C = 56.3 units A = 153.9 units2, C = 44.0 units A = 28.3 units2, C = 18.8 units A = 113.0 units2, C = 37.7 units 11) If the diameter of the average automobile tire is 2 ft, about how many revolutions does the wheel make for every mile driven? (Hint: 1 mi = 5280 ft.) a) b) c) d) 720 640 840 632